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Theorem bnj150 30200
 Description: Technical lemma for bnj151 30201. 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
bnj150.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj150.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj150.3 (𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
bnj150.4 (𝜑′[1𝑜 / 𝑛]𝜑)
bnj150.5 (𝜓′[1𝑜 / 𝑛]𝜓)
bnj150.6 (𝜃0 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
bnj150.7 (𝜁′[1𝑜 / 𝑛]𝜁)
bnj150.8 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
bnj150.9 (𝜑″[𝐹 / 𝑓]𝜑′)
bnj150.10 (𝜓″[𝐹 / 𝑓]𝜓′)
bnj150.11 (𝜁″[𝐹 / 𝑓]𝜁′)
Assertion
Ref Expression
bnj150 𝜃0
Distinct variable groups:   𝐴,𝑓,𝑛,𝑥   𝑓,𝐹,𝑖,𝑦   𝑅,𝑓,𝑛,𝑥   𝑓,𝜁″   𝑖,𝑛,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜁(𝑥,𝑦,𝑓,𝑖,𝑛)   𝐴(𝑦,𝑖)   𝑅(𝑦,𝑖)   𝐹(𝑥,𝑛)   𝜑′(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜁′(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜑″(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓″(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜁″(𝑥,𝑦,𝑖,𝑛)   𝜃0(𝑥,𝑦,𝑓,𝑖,𝑛)

Proof of Theorem bnj150
StepHypRef Expression
1 0ex 4718 . . . . . . . . . 10 ∅ ∈ V
2 bnj93 30187 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
3 funsng 5851 . . . . . . . . . 10 ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
41, 2, 3sylancr 694 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑥𝐴) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
5 bnj150.8 . . . . . . . . . 10 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
65funeqi 5824 . . . . . . . . 9 (Fun 𝐹 ↔ Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
74, 6sylibr 223 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → Fun 𝐹)
85bnj96 30189 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → dom 𝐹 = 1𝑜)
97, 8bnj1422 30162 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴) → 𝐹 Fn 1𝑜)
105bnj97 30190 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
11 bnj150.1 . . . . . . . . 9 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
12 bnj150.4 . . . . . . . . 9 (𝜑′[1𝑜 / 𝑛]𝜑)
13 bnj150.9 . . . . . . . . 9 (𝜑″[𝐹 / 𝑓]𝜑′)
1411, 12, 13, 5bnj125 30196 . . . . . . . 8 (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
1510, 14sylibr 223 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴) → 𝜑″)
169, 15jca 553 . . . . . 6 ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″))
17 bnj98 30191 . . . . . . 7 𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
18 bnj150.2 . . . . . . . 8 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
19 bnj150.5 . . . . . . . 8 (𝜓′[1𝑜 / 𝑛]𝜓)
20 bnj150.10 . . . . . . . 8 (𝜓″[𝐹 / 𝑓]𝜓′)
2118, 19, 20, 5bnj126 30197 . . . . . . 7 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
2217, 21mpbir 220 . . . . . 6 𝜓″
2316, 22jctir 559 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴) → ((𝐹 Fn 1𝑜𝜑″) ∧ 𝜓″))
24 df-3an 1033 . . . . 5 ((𝐹 Fn 1𝑜𝜑″𝜓″) ↔ ((𝐹 Fn 1𝑜𝜑″) ∧ 𝜓″))
2523, 24sylibr 223 . . . 4 ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″𝜓″))
26 bnj150.11 . . . . 5 (𝜁″[𝐹 / 𝑓]𝜁′)
27 bnj150.3 . . . . . 6 (𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
28 bnj150.7 . . . . . 6 (𝜁′[1𝑜 / 𝑛]𝜁)
2927, 28, 12, 19bnj121 30194 . . . . 5 (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)))
305, 13, 20, 26, 29bnj124 30195 . . . 4 (𝜁″ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″𝜓″)))
3125, 30mpbir 220 . . 3 𝜁″
325bnj95 30188 . . . 4 𝐹 ∈ V
33 sbceq1a 3413 . . . . 5 (𝑓 = 𝐹 → (𝜁′[𝐹 / 𝑓]𝜁′))
3433, 26syl6bbr 277 . . . 4 (𝑓 = 𝐹 → (𝜁′𝜁″))
3532, 34spcev 3273 . . 3 (𝜁″ → ∃𝑓𝜁′)
3631, 35ax-mp 5 . 2 𝑓𝜁′
37 bnj150.6 . . . 4 (𝜃0 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
38 19.37v 1897 . . . 4 (∃𝑓((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
3937, 38bitr4i 266 . . 3 (𝜃0 ↔ ∃𝑓((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)))
4039, 29bnj133 30047 . 2 (𝜃0 ↔ ∃𝑓𝜁′)
4136, 40mpbir 220 1 𝜃0
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ ciun 4455  suc csuc 5642  Fun wfun 5798   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-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-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-1o 7447  df-bnj13 30010  df-bnj15 30012 This theorem is referenced by:  bnj151  30201
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