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Theorem bnj944 32481
Description: Technical lemma for bnj69 32553. 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 486 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → (𝑅 FrSe 𝐴𝑋𝐴))
2 bnj944.3 . . . . . . . 8 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
3 bnj667 32294 . . . . . . . 8 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) → (𝑓 Fn 𝑛𝜑𝜓))
42, 3sylbi 220 . . . . . . 7 (𝜒 → (𝑓 Fn 𝑛𝜑𝜓))
5 bnj944.14 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))
64, 5sylibr 237 . . . . . 6 (𝜒𝜏)
763ad2ant1 1134 . . . . 5 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝜏)
87adantl 485 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜏)
92bnj1232 32346 . . . . . . 7 (𝜒𝑛𝐷)
10 vex 3401 . . . . . . . 8 𝑚 ∈ V
1110bnj216 32273 . . . . . . 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 278 . . . . . 6 (𝜎 ↔ (𝑛𝐷𝑚𝑛𝑝 = suc 𝑛))
1713, 16sylibr 237 . . . . 5 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝜎)
1817adantl 485 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜎)
19 bnj253 32245 . . . 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 32480 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅) ∈ V)
2621, 25eqeltrid 2837 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝐶 ∈ V)
2720, 26syl 17 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
28 bnj658 32293 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) → (𝑛𝐷𝑓 Fn 𝑛𝜑))
292, 28sylbi 220 . . . . 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 32252 . . . 4 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ ((𝑛𝐷𝑓 Fn 𝑛𝜑) ∧ 𝑝 = suc 𝑛))
3330, 31, 32sylanbrc 586 . . 3 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑))
3433adantl 485 . 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 4758 . . . . . . . . 9 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ⟨𝑛, 𝐶⟩ = ⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩)
3837sneqd 4525 . . . . . . . 8 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {⟨𝑛, 𝐶⟩} = {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
3938uneq2d 4051 . . . . . . 7 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
4036, 39syl5eq 2785 . . . . . 6 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
4140sbceq1d 3684 . . . . 5 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ([𝐺 / 𝑓]𝜑′[(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′))
4235, 41syl5bb 286 . . . 4 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝜑″[(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′))
4342imbi2d 344 . . 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 264 . . . 4 ([(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′[(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)
46 eqid 2738 . . . 4 (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
47 0ex 5172 . . . . 5 ∅ ∈ V
4847elimel 4480 . . . 4 if(𝐶 ∈ V, 𝐶, ∅) ∈ V
4923, 44, 45, 22, 46, 48bnj929 32479 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)
5043, 49dedth 4469 . 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2113  wral 3053  Vcvv 3397  [wsbc 3679  cdif 3838  cun 3839  c0 4209  ifcif 4411  {csn 4513  cop 4519   ciun 4878  suc csuc 6168   Fn wfn 6328  cfv 6333  ωcom 7593  w-bnj17 32227   predc-bnj14 32229   FrSe w-bnj15 32233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473  ax-reg 9122
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-om 7594  df-bnj17 32228  df-bnj14 32230  df-bnj13 32232  df-bnj15 32234
This theorem is referenced by:  bnj910  32491
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