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Theorem bnj944 30716
Description: Technical lemma for bnj69 30786. 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
bnj944.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj944.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj944.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj944.4 (𝜑′[𝑝 / 𝑛]𝜑)
bnj944.7 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj944.10 𝐷 = (ω ∖ {∅})
bnj944.12 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj944.13 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj944.14 (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))
bnj944.15 (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
Assertion
Ref Expression
bnj944 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜑″)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛   𝑦,𝐴,𝑓,𝑖,𝑚   𝑅,𝑓,𝑖,𝑚,𝑛   𝑦,𝑅   𝑓,𝑋,𝑛
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜏(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑝)   𝐺(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑖,𝑚,𝑝)   𝜑′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj944
StepHypRef Expression
1 simpl 473 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → (𝑅 FrSe 𝐴𝑋𝐴))
2 bnj944.3 . . . . . . . 8 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
3 bnj667 30530 . . . . . . . 8 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) → (𝑓 Fn 𝑛𝜑𝜓))
42, 3sylbi 207 . . . . . . 7 (𝜒 → (𝑓 Fn 𝑛𝜑𝜓))
5 bnj944.14 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))
64, 5sylibr 224 . . . . . 6 (𝜒𝜏)
763ad2ant1 1080 . . . . 5 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝜏)
87adantl 482 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜏)
92bnj1232 30582 . . . . . . 7 (𝜒𝑛𝐷)
10 vex 3189 . . . . . . . 8 𝑚 ∈ V
1110bnj216 30508 . . . . . . 7 (𝑛 = suc 𝑚𝑚𝑛)
12 id 22 . . . . . . 7 (𝑝 = suc 𝑛𝑝 = suc 𝑛)
139, 11, 123anim123i 1245 . . . . . 6 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛𝐷𝑚𝑛𝑝 = suc 𝑛))
14 bnj944.15 . . . . . . 7 (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
15 3ancomb 1045 . . . . . . 7 ((𝑛𝐷𝑝 = suc 𝑛𝑚𝑛) ↔ (𝑛𝐷𝑚𝑛𝑝 = suc 𝑛))
1614, 15bitri 264 . . . . . 6 (𝜎 ↔ (𝑛𝐷𝑚𝑛𝑝 = suc 𝑛))
1713, 16sylibr 224 . . . . 5 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝜎)
1817adantl 482 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜎)
19 bnj253 30477 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜏𝜎))
201, 8, 18, 19syl3anbrc 1244 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → (𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎))
21 bnj944.12 . . . 4 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
22 bnj944.10 . . . . 5 𝐷 = (ω ∖ {∅})
23 bnj944.1 . . . . 5 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
24 bnj944.2 . . . . 5 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
2522, 5, 14, 23, 24bnj938 30715 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅) ∈ V)
2621, 25syl5eqel 2702 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝐶 ∈ V)
2720, 26syl 17 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
28 bnj658 30529 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) → (𝑛𝐷𝑓 Fn 𝑛𝜑))
292, 28sylbi 207 . . . . 5 (𝜒 → (𝑛𝐷𝑓 Fn 𝑛𝜑))
30293ad2ant1 1080 . . . 4 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛𝐷𝑓 Fn 𝑛𝜑))
31 simp3 1061 . . . 4 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑝 = suc 𝑛)
32 bnj291 30484 . . . 4 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ ((𝑛𝐷𝑓 Fn 𝑛𝜑) ∧ 𝑝 = suc 𝑛))
3330, 31, 32sylanbrc 697 . . 3 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑))
3433adantl 482 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → (𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑))
35 bnj944.7 . . . . 5 (𝜑″[𝐺 / 𝑓]𝜑′)
36 bnj944.13 . . . . . . 7 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
37 opeq2 4371 . . . . . . . . 9 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ⟨𝑛, 𝐶⟩ = ⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩)
3837sneqd 4160 . . . . . . . 8 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {⟨𝑛, 𝐶⟩} = {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
3938uneq2d 3745 . . . . . . 7 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
4036, 39syl5eq 2667 . . . . . 6 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
4140sbceq1d 3422 . . . . 5 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ([𝐺 / 𝑓]𝜑′[(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′))
4235, 41syl5bb 272 . . . 4 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝜑″[(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′))
4342imbi2d 330 . . 3 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝜑″) ↔ ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)))
44 bnj944.4 . . . 4 (𝜑′[𝑝 / 𝑛]𝜑)
45 biid 251 . . . 4 ([(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′[(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)
46 eqid 2621 . . . 4 (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
47 0ex 4750 . . . . 5 ∅ ∈ V
4847elimel 4122 . . . 4 if(𝐶 ∈ V, 𝐶, ∅) ∈ V
4923, 44, 45, 22, 46, 48bnj929 30714 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)
5043, 49dedth 4111 . 2 (𝐶 ∈ V → ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝜑″))
5127, 34, 50sylc 65 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜑″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  [wsbc 3417  cdif 3552  cun 3553  c0 3891  ifcif 4058  {csn 4148  cop 4154   ciun 4485  suc csuc 5684   Fn wfn 5842  cfv 5847  ωcom 7012  w-bnj17 30459   predc-bnj14 30461   FrSe w-bnj15 30465
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 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902  ax-reg 8441
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 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-bnj17 30460  df-bnj14 30462  df-bnj13 30464  df-bnj15 30466
This theorem is referenced by:  bnj910  30726
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