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Theorem bnj944 33949
Description: Technical lemma for bnj69 34021. 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 484 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → (𝑅 FrSe 𝐴𝑋𝐴))
2 bnj944.3 . . . . . . . 8 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
3 bnj667 33763 . . . . . . . 8 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) → (𝑓 Fn 𝑛𝜑𝜓))
42, 3sylbi 216 . . . . . . 7 (𝜒 → (𝑓 Fn 𝑛𝜑𝜓))
5 bnj944.14 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))
64, 5sylibr 233 . . . . . 6 (𝜒𝜏)
763ad2ant1 1134 . . . . 5 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝜏)
87adantl 483 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜏)
92bnj1232 33814 . . . . . . 7 (𝜒𝑛𝐷)
10 vex 3479 . . . . . . . 8 𝑚 ∈ V
1110bnj216 33743 . . . . . . 7 (𝑛 = suc 𝑚𝑚𝑛)
12 id 22 . . . . . . 7 (𝑝 = suc 𝑛𝑝 = suc 𝑛)
139, 11, 123anim123i 1152 . . . . . 6 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛𝐷𝑚𝑛𝑝 = suc 𝑛))
14 bnj944.15 . . . . . . 7 (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
15 3ancomb 1100 . . . . . . 7 ((𝑛𝐷𝑝 = suc 𝑛𝑚𝑛) ↔ (𝑛𝐷𝑚𝑛𝑝 = suc 𝑛))
1614, 15bitri 275 . . . . . 6 (𝜎 ↔ (𝑛𝐷𝑚𝑛𝑝 = suc 𝑛))
1713, 16sylibr 233 . . . . 5 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝜎)
1817adantl 483 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜎)
19 bnj253 33715 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜏𝜎))
201, 8, 18, 19syl3anbrc 1344 . . 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 33948 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅) ∈ V)
2621, 25eqeltrid 2838 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝐶 ∈ V)
2720, 26syl 17 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
28 bnj658 33762 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) → (𝑛𝐷𝑓 Fn 𝑛𝜑))
292, 28sylbi 216 . . . . 5 (𝜒 → (𝑛𝐷𝑓 Fn 𝑛𝜑))
30293ad2ant1 1134 . . . 4 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛𝐷𝑓 Fn 𝑛𝜑))
31 simp3 1139 . . . 4 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑝 = suc 𝑛)
32 bnj291 33722 . . . 4 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ ((𝑛𝐷𝑓 Fn 𝑛𝜑) ∧ 𝑝 = suc 𝑛))
3330, 31, 32sylanbrc 584 . . 3 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑))
3433adantl 483 . 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 4875 . . . . . . . . 9 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ⟨𝑛, 𝐶⟩ = ⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩)
3837sneqd 4641 . . . . . . . 8 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {⟨𝑛, 𝐶⟩} = {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
3938uneq2d 4164 . . . . . . 7 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
4036, 39eqtrid 2785 . . . . . 6 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
4140sbceq1d 3783 . . . . 5 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ([𝐺 / 𝑓]𝜑′[(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′))
4235, 41bitrid 283 . . . 4 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝜑″[(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′))
4342imbi2d 341 . . 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 261 . . . 4 ([(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′[(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)
46 eqid 2733 . . . 4 (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
47 0ex 5308 . . . . 5 ∅ ∈ V
4847elimel 4598 . . . 4 if(𝐶 ∈ V, 𝐶, ∅) ∈ V
4923, 44, 45, 22, 46, 48bnj929 33947 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)
5043, 49dedth 4587 . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  [wsbc 3778  cdif 3946  cun 3947  c0 4323  ifcif 4529  {csn 4629  cop 4635   ciun 4998  suc csuc 6367   Fn wfn 6539  cfv 6544  ωcom 7855  w-bnj17 33697   predc-bnj14 33699   FrSe w-bnj15 33703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-reg 9587
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-om 7856  df-bnj17 33698  df-bnj14 33700  df-bnj13 33702  df-bnj15 33704
This theorem is referenced by:  bnj910  33959
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