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Theorem List for Metamath Proof Explorer - 30201-30300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembnj151 30201* Technical lemma for bnj153 30204. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜏 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜃))    &   (𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜑′[1𝑜 / 𝑛]𝜑)    &   (𝜓′[1𝑜 / 𝑛]𝜓)    &   (𝜃′[1𝑜 / 𝑛]𝜃)    &   (𝜃0 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))    &   (𝜃1 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))    &   (𝜁′[1𝑜 / 𝑛]𝜁)    &   𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}    &   (𝜑″[𝐹 / 𝑓]𝜑′)    &   (𝜓″[𝐹 / 𝑓]𝜓′)    &   (𝜁″[𝐹 / 𝑓]𝜁′)    &   (𝜁0 ↔ (𝑓 Fn 1𝑜𝜑′𝜓′))    &   (𝜁1[𝑔 / 𝑓]𝜁0)    &   (𝜑1[𝑔 / 𝑓]𝜑′)    &   (𝜓1[𝑔 / 𝑓]𝜓′)       (𝑛 = 1𝑜 → ((𝑛𝐷𝜏) → 𝜃))

Theorembnj154 30202* Technical lemma for bnj153 30204. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑1[𝑔 / 𝑓]𝜑′)    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))       (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))

Theorembnj155 30203* Technical lemma for bnj153 30204. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓1[𝑔 / 𝑓]𝜓′)    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       (𝜓1 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj153 30204* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜏 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜃))       (𝑛 = 1𝑜 → ((𝑛𝐷𝜏) → 𝜃))

Theorembnj207 30205* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜑′[𝑀 / 𝑛]𝜑)    &   (𝜓′[𝑀 / 𝑛]𝜓)    &   (𝜒′[𝑀 / 𝑛]𝜒)    &   𝑀 ∈ V       (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑀𝜑′𝜓′)))

Theorembnj213 30206 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴

Theorembnj222 30207* Technical lemma for bnj229 30208. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))       (𝜓 ↔ ∀𝑚 ∈ ω (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))

Theorembnj229 30208* Technical lemma for bnj517 30209. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑛𝑁 ∧ (suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓)) → (𝐹𝑛) ⊆ 𝐴)

Theorembnj517 30209* Technical lemma for bnj518 30210. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
(𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑁 ∈ ω ∧ 𝜑𝜓) → ∀𝑛𝑁 (𝐹𝑛) ⊆ 𝐴)

Theorembnj518 30210* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝜑𝜓𝑛 ∈ ω ∧ 𝑝𝑛))       ((𝑅 FrSe 𝐴𝜏) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Theorembnj523 30211* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜑′[𝑀 / 𝑛]𝜑)    &   𝑀 ∈ V       (𝜑′ ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))

Theorembnj526 30212* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜑″[𝐺 / 𝑓]𝜑)    &   𝐺 ∈ V       (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

Theorembnj528 30213 Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})       𝐺 ∈ V

Theorembnj535 30214* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝜑′𝜓′𝑚 ∈ ω ∧ 𝑝𝑚))       ((𝑅 FrSe 𝐴𝜏𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)

Theorembnj539 30215* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓′[𝑀 / 𝑛]𝜓)    &   𝑀 ∈ V       (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑀 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj540 30216* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   𝐺 ∈ V       (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj543 30217* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝𝑚))       ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)

Theorembnj544 30218* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))       ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)

Theorembnj545 30219 Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   𝐷 = (ω ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)    &   (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))       ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)

Theorembnj546 30220* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑅 FrSe 𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Theorembnj548 30221* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)       (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → 𝐵 = 𝐾)

Theorembnj553 30222* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)       (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚𝑝 = 𝑖) → (𝐺𝑚) = 𝐿)

Theorembnj554 30223* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)       ((𝜂𝜁) → ((𝐺𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾))

Theorembnj556 30224 Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))       (𝜂𝜎)

Theorembnj557 30225* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)       ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → (𝐺𝑚) = 𝐿)

Theorembnj558 30226* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)       ((𝑅 FrSe 𝐴𝜏𝜂𝜁) → (𝐺‘suc 𝑖) = 𝐾)

Theorembnj561 30227 Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)       ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)

Theorembnj562 30228 Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)       ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜑″)

Theorembnj570 30229* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑅 FrSe 𝐴𝜏𝜂𝜌) → (𝐺‘suc 𝑖) = 𝐾)

Theorembnj571 30230* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)    &   (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)    &   (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜓″)

Theorembnj605 30231* Technical lemma. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))    &   (𝜑″[𝑓 / 𝑓]𝜑)    &   (𝜓″[𝑓 / 𝑓]𝜓)    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   𝑓 ∈ V    &   (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))    &   (𝜑″ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚𝑝𝜂)    &   ((𝜃𝑚𝐷𝑚 E 𝑛) → 𝜒′)    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝑓 Fn 𝑛)    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜑″)    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜓″)       ((𝑛 ≠ 1𝑜𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 𝑛𝜑𝜓)))

Theorembnj581 30232* Technical lemma for bnj580 30237. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 9-Jul-2011.) (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   (𝜑′[𝑔 / 𝑓]𝜑)    &   (𝜓′[𝑔 / 𝑓]𝜓)    &   (𝜒′[𝑔 / 𝑓]𝜒)       (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))

Theorembnj589 30233* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       (𝜓 ↔ ∀𝑘 ∈ ω (suc 𝑘𝑛 → (𝑓‘suc 𝑘) = 𝑦 ∈ (𝑓𝑘) pred(𝑦, 𝐴, 𝑅)))

Theorembnj590 30234 Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝐵 = suc 𝑖𝜓) → (𝑖 ∈ ω → (𝐵𝑛 → (𝑓𝐵) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))

Theorembnj591 30235* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))       ([𝑘 / 𝑗]𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))

Theorembnj594 30236* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})    &   (𝜑′ ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))    &   (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))    &   ([𝑘 / 𝑗]𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))    &   (𝜏 ↔ ∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]𝜃))       ((𝑗𝑛𝜏) → 𝜃)

Theorembnj580 30237* Technical lemma for bnj579 30238. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   (𝜑′[𝑔 / 𝑓]𝜑)    &   (𝜓′[𝑔 / 𝑓]𝜓)    &   (𝜒′[𝑔 / 𝑓]𝜒)    &   𝐷 = (ω ∖ {∅})    &   (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))    &   (𝜏 ↔ ∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]𝜃))       (𝑛𝐷 → ∃*𝑓𝜒)

Theorembnj579 30238* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})       (𝑛𝐷 → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))

Theorembnj602 30239 Equality theorem for the pred function constant. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))

Theorembnj607 30240* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))    &   (𝜑″[𝐺 / 𝑓]𝜑)    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   𝐺 ∈ V    &   (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))    &   (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚𝑝𝜂)    &   ((𝜃𝑚𝐷𝑚 E 𝑛) → 𝜒′)    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜑″)    &   ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜓″)    &   (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜑0[ / 𝑓]𝜑)    &   (𝜓0[ / 𝑓]𝜓)    &   (𝜑1[𝐺 / ]𝜑0)    &   (𝜓1[𝐺 / ]𝜓0)       ((𝑛 ≠ 1𝑜𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 𝑛𝜑𝜓)))

Theorembnj609 30241* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜑″[𝐺 / 𝑓]𝜑)    &   𝐺 ∈ V       (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

Theorembnj611 30242* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   𝐺 ∈ V       (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj600 30243* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))    &   (𝜑′[𝑚 / 𝑛]𝜑)    &   (𝜓′[𝑚 / 𝑛]𝜓)    &   (𝜒′[𝑚 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑)    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   (𝜒″[𝐺 / 𝑓]𝜒)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))    &   (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})       (𝑛 ≠ 1𝑜 → ((𝑛𝐷𝜃) → 𝜒))

Theorembnj601 30244* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))       (𝑛 ≠ 1𝑜 → ((𝑛𝐷𝜃) → 𝜒))

Theorembnj852 30245* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})       ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))

Theorembnj864 30246* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))    &   (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))       (𝜒 → ∃!𝑓𝜃)

Theorembnj865 30247* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))    &   (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))       𝑤𝑛(𝜒 → ∃𝑓𝑤 𝜃)

Theorembnj873 30248* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜑′[𝑔 / 𝑓]𝜑)    &   (𝜓′[𝑔 / 𝑓]𝜓)       𝐵 = {𝑔 ∣ ∃𝑛𝐷 (𝑔 Fn 𝑛𝜑′𝜓′)}

Theorembnj849 30249* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑛𝐷))    &   (𝜃 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   (𝜑′[𝑔 / 𝑓]𝜑)    &   (𝜓′[𝑔 / 𝑓]𝜓)    &   (𝜃′[𝑔 / 𝑓]𝜃)    &   (𝜏 ↔ (𝑅 FrSe 𝐴𝑋𝐴))       ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ∈ V)

Theorembnj882 30250* Definition (using hypotheses for readability) of the function giving the transitive closure of 𝑋 in 𝐴 by 𝑅. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}        trCl(𝑋, 𝐴, 𝑅) = 𝑓𝐵 𝑖 ∈ dom 𝑓(𝑓𝑖)

Theorembnj18eq1 30251 Equality theorem for transitive closure. (Contributed by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
(𝑋 = 𝑌 → trCl(𝑋, 𝐴, 𝑅) = trCl(𝑌, 𝐴, 𝑅))

Theorembnj893 30252 Property of trCl. Under certain conditions, the transitive closure of 𝑋 in 𝐴 by 𝑅 is a set. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)

Theorembnj900 30253* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       (𝑓𝐵 → ∅ ∈ dom 𝑓)

Theorembnj906 30254 Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))

Theorembnj908 30255* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))    &   (𝜑′[𝑚 / 𝑛]𝜑)    &   (𝜓′[𝑚 / 𝑛]𝜓)    &   (𝜒′[𝑚 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑)    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   (𝜒″[𝐺 / 𝑓]𝜒)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))    &   (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})       ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓(𝐺 Fn 𝑛𝜑″𝜓″))

Theorembnj911 30256* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑓 Fn 𝑛𝜑𝜓) → ∀𝑖(𝑓 Fn 𝑛𝜑𝜓))

Theorembnj916 30257* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))       (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝑛𝐷𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))

Theorembnj917 30258* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))       (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))

Theorembnj934 30259* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   𝐺 ∈ V       ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)

Theorembnj929 30260* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   𝐷 = (ω ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   𝐶 ∈ V       ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝜑″)

Theorembnj938 30261* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Theorembnj944 30262* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   𝐷 = (ω ∖ {∅})    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜑″)

Theorembnj953 30263 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))       (((𝐺𝑖) = (𝑓𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))

Theorembnj958 30264* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))

Theorembnj1000 30265* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   𝐺 ∈ V    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj965 30266* Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj964 30267* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))    &   (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜓″)

Theorembnj966 30268* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)    &   (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐺 Fn 𝑝)       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))

Theorembnj967 30269* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))

Theorembnj969 30270* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)

Theorembnj970 30271 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝𝐷)

Theorembnj910 30272* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜒″)

Theorembnj978 30273* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜃𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))       ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))

Theorembnj981 30274* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))       (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)))

Theorembnj983 30275* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))       (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)))

Theorembnj984 30276 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       (𝐺𝐴 → (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒))

Theorembnj985 30277* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       (𝐺𝐵 ↔ ∃𝑝𝜒″)

Theorembnj986 30278* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))       (𝜒 → ∃𝑚𝑝𝜏)

Theorembnj996 30279* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂))

Theorembnj998 30280* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       ((𝜃𝜒𝜏𝜂) → 𝜒″)

Theorembnj999 30281* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))

Theorembnj1001 30282 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   ((𝜃𝜒𝜏𝜂) → 𝜒″)       ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))

Theorembnj1006 30283* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐷 = (ω ∖ {∅})    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))       ((𝜃𝜒𝜏𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))

Theorembnj1014 30284* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       ((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))

Theorembnj1015 30285* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐺𝑉    &   𝐽𝑉       ((𝐺𝐵𝐽 ∈ dom 𝐺) → (𝐺𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅))

Theorembnj1018 30286* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))    &   ((𝜃𝜒𝜏𝜂) → 𝜒″)    &   ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))       ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))

Theorembnj1020 30287* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))       ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))

Theorembnj1021 30288* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))

Theorembnj907 30289* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))

Theorembnj1029 30290 Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 FrSe 𝐴𝑋𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))

Theorembnj1033 30291* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   (𝜂𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))    &   (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵)       ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

Theorembnj1034 30292* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵)       ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

Theorembnj1039 30293 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓′[𝑗 / 𝑖]𝜓)       (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))

Theorembnj1040 30294* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑′[𝑗 / 𝑖]𝜑)    &   (𝜓′[𝑗 / 𝑖]𝜓)    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜒′[𝑗 / 𝑖]𝜒)       (𝜒′ ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′))

Theorembnj1047 30295 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))    &   (𝜂′[𝑗 / 𝑖]𝜂)       (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖𝜂′))

Theorembnj1049 30296 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   (𝜂 ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))       (∀𝑖𝑛 𝜂 ↔ ∀𝑖𝜂)

Theorembnj1052 30297* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜂 ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))    &   (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))    &   ((𝜃𝜏𝜒𝜁) → ( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)))       ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

Theorembnj1053 30298* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜂 ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))    &   (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))    &   ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 (𝜌𝜂))       ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

Theorembnj1071 30299 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})       (𝑛𝐷 → E Fr 𝑛)

Theorembnj1083 30300 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       (𝑓𝐾 ↔ ∃𝑛𝜒)

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