Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj153 Structured version   Visualization version   GIF version

Theorem bnj153 30204
Description: 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.)
Hypotheses
Ref Expression
bnj153.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj153.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj153.3 𝐷 = (ω ∖ {∅})
bnj153.4 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
bnj153.5 (𝜏 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜃))
Assertion
Ref Expression
bnj153 (𝑛 = 1𝑜 → ((𝑛𝐷𝜏) → 𝜃))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑥,𝑦,𝑛   𝑅,𝑓,𝑖,𝑥,𝑦,𝑛   𝑚,𝑛
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜃(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜏(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝐴(𝑚)   𝐷(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝑅(𝑚)

Proof of Theorem bnj153
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 bnj153.1 . 2 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2 bnj153.2 . 2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj153.3 . 2 𝐷 = (ω ∖ {∅})
4 bnj153.4 . 2 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
5 bnj153.5 . 2 (𝜏 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜃))
6 biid 250 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
7 biid 250 . . . 4 ([1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜑)
81, 7bnj118 30193 . . 3 ([1𝑜 / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
98bicomi 213 . 2 ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ [1𝑜 / 𝑛]𝜑)
10 bnj105 30044 . . . 4 1𝑜 ∈ V
112, 10bnj92 30186 . . 3 ([1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
1211bicomi 213 . 2 (∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [1𝑜 / 𝑛]𝜓)
13 biid 250 . 2 ([1𝑜 / 𝑛]𝜃[1𝑜 / 𝑛]𝜃)
14 biid 250 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
15 biid 250 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
16 biid 250 . . . . 5 ([1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ [1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
17 biid 250 . . . . 5 ([1𝑜 / 𝑛]𝜓[1𝑜 / 𝑛]𝜓)
186, 16, 7, 17bnj121 30194 . . . 4 ([1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓)))
198anbi2i 726 . . . . . . 7 ((𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑) ↔ (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)))
2019, 11anbi12i 729 . . . . . 6 (((𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑) ∧ [1𝑜 / 𝑛]𝜓) ↔ ((𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
21 df-3an 1033 . . . . . 6 ((𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ ((𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑) ∧ [1𝑜 / 𝑛]𝜓))
22 df-3an 1033 . . . . . 6 ((𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ((𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
2320, 21, 223bitr4i 291 . . . . 5 ((𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
2423imbi2i 325 . . . 4 (((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
2518, 24bitri 263 . . 3 ([1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
2625bicomi 213 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ [1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
27 eqid 2610 . 2 {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
28 biid 250 . 2 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
29 biid 250 . 2 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3026sbcbii 3458 . . 3 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
31 biid 250 . . . . 5 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑)
32 biid 250 . . . . 5 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓)
33 biid 250 . . . . 5 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
3427, 31, 32, 33, 18bnj124 30195 . . . 4 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓)))
351, 7, 31, 27bnj125 30196 . . . . . . . 8 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑 ↔ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅))
3635anbi2i 726 . . . . . . 7 (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑) ↔ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅)))
372, 17, 32, 27bnj126 30197 . . . . . . 7 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3836, 37anbi12i 729 . . . . . 6 ((({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑) ∧ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓) ↔ (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅))))
39 df-3an 1033 . . . . . 6 (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓) ↔ (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑) ∧ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓))
40 df-3an 1033 . . . . . 6 (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅))))
4138, 39, 403bitr4i 291 . . . . 5 (({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓) ↔ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅))))
4241imbi2i 325 . . . 4 (((𝑅 FrSe 𝐴𝑥𝐴) → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜑[{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅)))))
4334, 42bitri 263 . . 3 ([{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓][1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅)))))
4430, 43bitr2i 264 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} Fn 1𝑜 ∧ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘suc 𝑖) = 𝑦 ∈ ({⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}‘𝑖) pred(𝑦, 𝐴, 𝑅)))) ↔ [{⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} / 𝑓]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
45 biid 250 . 2 ((𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
46 biid 250 . . . . 5 ((𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ (𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
47 biid 250 . . . . 5 ([𝑔 / 𝑓](𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ [𝑔 / 𝑓](𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
48 biid 250 . . . . 5 ([𝑔 / 𝑓][1𝑜 / 𝑛]𝜑[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑)
49 biid 250 . . . . 5 ([𝑔 / 𝑓][1𝑜 / 𝑛]𝜓[𝑔 / 𝑓][1𝑜 / 𝑛]𝜓)
5046, 47, 48, 49bnj156 30050 . . . 4 ([𝑔 / 𝑓](𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ (𝑔 Fn 1𝑜[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑[𝑔 / 𝑓][1𝑜 / 𝑛]𝜓))
5148, 8bnj154 30202 . . . . . . 7 ([𝑔 / 𝑓][1𝑜 / 𝑛]𝜑 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
5251anbi2i 726 . . . . . 6 ((𝑔 Fn 1𝑜[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑) ↔ (𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)))
5317, 11bitri 263 . . . . . . 7 ([1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
5449, 53bnj155 30203 . . . . . 6 ([𝑔 / 𝑓][1𝑜 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
5552, 54anbi12i 729 . . . . 5 (((𝑔 Fn 1𝑜[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑) ∧ [𝑔 / 𝑓][1𝑜 / 𝑛]𝜓) ↔ ((𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))))
56 df-3an 1033 . . . . 5 ((𝑔 Fn 1𝑜[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑[𝑔 / 𝑓][1𝑜 / 𝑛]𝜓) ↔ ((𝑔 Fn 1𝑜[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑) ∧ [𝑔 / 𝑓][1𝑜 / 𝑛]𝜓))
57 df-3an 1033 . . . . 5 ((𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ ((𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))))
5855, 56, 573bitr4i 291 . . . 4 ((𝑔 Fn 1𝑜[𝑔 / 𝑓][1𝑜 / 𝑛]𝜑[𝑔 / 𝑓][1𝑜 / 𝑛]𝜓) ↔ (𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))))
5950, 58bitri 263 . . 3 ([𝑔 / 𝑓](𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ (𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))))
6023sbcbii 3458 . . 3 ([𝑔 / 𝑓](𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓) ↔ [𝑔 / 𝑓](𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
6159, 60bitr3i 265 . 2 ((𝑔 Fn 1𝑜 ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ [𝑔 / 𝑓](𝑓 Fn 1𝑜 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
62 biid 250 . 2 ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ [𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
63 biid 250 . 2 ([𝑔 / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ [𝑔 / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
641, 2, 3, 4, 5, 6, 9, 12, 13, 14, 15, 26, 27, 28, 29, 44, 45, 61, 62, 63bnj151 30201 1 (𝑛 = 1𝑜 → ((𝑛𝐷𝜏) → 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wex 1695  wcel 1977  ∃!weu 2458  ∃*wmo 2459  wral 2896  [wsbc 3402  cdif 3537  c0 3874  {csn 4125  cop 4131   ciun 4455   class class class wbr 4583   E cep 4947  suc csuc 5642   Fn wfn 5799  cfv 5804  ωcom 6957  1𝑜c1o 7440   predc-bnj14 30007   FrSe w-bnj15 30011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-bnj13 30010  df-bnj15 30012
This theorem is referenced by:  bnj852  30245
  Copyright terms: Public domain W3C validator