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Theorem bnj910 30718
Description: Technical lemma for bnj69 30778. 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
bnj910.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj910.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj910.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj910.4 (𝜑′[𝑝 / 𝑛]𝜑)
bnj910.5 (𝜓′[𝑝 / 𝑛]𝜓)
bnj910.6 (𝜒′[𝑝 / 𝑛]𝜒)
bnj910.7 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj910.8 (𝜓″[𝐺 / 𝑓]𝜓′)
bnj910.9 (𝜒″[𝐺 / 𝑓]𝜒′)
bnj910.10 𝐷 = (ω ∖ {∅})
bnj910.11 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj910.12 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj910.13 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj910.14 (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))
bnj910.15 (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
Assertion
Ref Expression
bnj910 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜒″)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐷,𝑓,𝑖,𝑛   𝑖,𝐺   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛   𝑓,𝑝,𝑖,𝑛   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑦,𝑓,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)   𝐵(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑚,𝑝)   𝑅(𝑝)   𝐺(𝑦,𝑓,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑚,𝑝)   𝜑′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒″(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj910
StepHypRef Expression
1 bnj910.3 . . . 4 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
2 bnj910.10 . . . 4 𝐷 = (ω ∖ {∅})
31, 2bnj970 30717 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝𝐷)
4 bnj910.1 . . . . 5 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
5 bnj910.2 . . . . 5 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
6 bnj910.12 . . . . 5 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
7 bnj910.14 . . . . 5 (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))
8 bnj910.15 . . . . 5 (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
94, 5, 1, 2, 6, 7, 8bnj969 30716 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
10 simpr3 1067 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝 = suc 𝑛)
111bnj1235 30575 . . . . . 6 (𝜒𝑓 Fn 𝑛)
12113ad2ant1 1080 . . . . 5 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑓 Fn 𝑛)
1312adantl 482 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑓 Fn 𝑛)
14 bnj910.13 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
1514bnj941 30543 . . . . 5 (𝐶 ∈ V → ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
16153impib 1259 . . . 4 ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
179, 10, 13, 16syl3anc 1323 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐺 Fn 𝑝)
18 bnj910.4 . . . 4 (𝜑′[𝑝 / 𝑛]𝜑)
19 bnj910.7 . . . 4 (𝜑″[𝐺 / 𝑓]𝜑′)
204, 5, 1, 18, 19, 2, 6, 14, 7, 8bnj944 30708 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜑″)
21 bnj910.5 . . . 4 (𝜓′[𝑝 / 𝑛]𝜓)
22 bnj910.8 . . . 4 (𝜓″[𝐺 / 𝑓]𝜓′)
235, 1, 2, 6, 14, 9bnj967 30715 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
241, 2, 6, 14, 9, 17bnj966 30714 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
255, 1, 21, 22, 6, 14, 23, 24bnj964 30713 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜓″)
263, 17, 20, 25bnj951 30546 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
27 bnj910.6 . . . 4 (𝜒′[𝑝 / 𝑛]𝜒)
28 vex 3194 . . . 4 𝑝 ∈ V
291, 18, 21, 27, 28bnj919 30537 . . 3 (𝜒′ ↔ (𝑝𝐷𝑓 Fn 𝑝𝜑′𝜓′))
30 bnj910.9 . . 3 (𝜒″[𝐺 / 𝑓]𝜒′)
3114bnj918 30536 . . 3 𝐺 ∈ V
3229, 19, 22, 30, 31bnj976 30548 . 2 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
3326, 32sylibr 224 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜒″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  {cab 2612  wral 2912  wrex 2913  Vcvv 3191  [wsbc 3422  cdif 3557  cun 3558  c0 3896  {csn 4153  cop 4159   ciun 4490  suc csuc 5687   Fn wfn 5845  cfv 5850  ωcom 7013  w-bnj17 30451   predc-bnj14 30453   FrSe w-bnj15 30457
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903  ax-reg 8442
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  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 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-bnj17 30452  df-bnj14 30454  df-bnj13 30456  df-bnj15 30458
This theorem is referenced by:  bnj998  30726
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