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Theorem bnj938 30750
 Description: Technical lemma for bnj69 30821. 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
bnj938.1 𝐷 = (ω ∖ {∅})
bnj938.2 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj938.3 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj938.4 (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj938.5 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj938 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑅,𝑖,𝑝,𝑦   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝
Allowed substitution hints:   𝜏(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑓,𝑚,𝑛)   𝐷(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑓,𝑚,𝑛)   𝑋(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj938
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elisset 3204 . . 3 (𝑋𝐴 → ∃𝑥 𝑥 = 𝑋)
21bnj706 30567 . 2 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → ∃𝑥 𝑥 = 𝑋)
3 bnj291 30519 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) ↔ ((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑋𝐴))
43simplbi 476 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → (𝑅 FrSe 𝐴𝜏𝜎))
5 bnj602 30728 . . . . . . . . . 10 (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
65eqeq2d 2631 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
7 bnj938.4 . . . . . . . . 9 (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
86, 7syl6bbr 278 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ 𝜑′))
983anbi2d 1401 . . . . . . 7 (𝑥 = 𝑋 → ((𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ↔ (𝑓 Fn 𝑚𝜑′𝜓′)))
10 bnj938.2 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
119, 10syl6bbr 278 . . . . . 6 (𝑥 = 𝑋 → ((𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ↔ 𝜏))
12113anbi2d 1401 . . . . 5 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ (𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ∧ 𝜎) ↔ (𝑅 FrSe 𝐴𝜏𝜎)))
134, 12syl5ibr 236 . . . 4 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → (𝑅 FrSe 𝐴 ∧ (𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ∧ 𝜎)))
14 bnj938.1 . . . . 5 𝐷 = (ω ∖ {∅})
15 biid 251 . . . . 5 ((𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ↔ (𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′))
16 bnj938.3 . . . . 5 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
17 biid 251 . . . . 5 ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
18 bnj938.5 . . . . 5 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
1914, 15, 16, 17, 18bnj546 30709 . . . 4 ((𝑅 FrSe 𝐴 ∧ (𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ∧ 𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
2013, 19syl6 35 . . 3 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V))
2120exlimiv 1855 . 2 (∃𝑥 𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V))
222, 21mpcom 38 1 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∀wral 2907  Vcvv 3189   ∖ cdif 3556  ∅c0 3896  {csn 4153  ∪ ciun 4490  suc csuc 5689   Fn wfn 5847  ‘cfv 5852  ωcom 7019   ∧ w-bnj17 30494   predc-bnj14 30496   FrSe w-bnj15 30500 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-bnj17 30495  df-bnj14 30497  df-bnj13 30499  df-bnj15 30501 This theorem is referenced by:  bnj944  30751  bnj969  30759
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