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Theorem bnj910 34707
Description: Technical lemma for bnj69 34769. 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 34706 . . 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 34705 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
10 simpr3 1193 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝 = suc 𝑛)
111bnj1235 34563 . . . . . 6 (𝜒𝑓 Fn 𝑛)
12113ad2ant1 1130 . . . . 5 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑓 Fn 𝑛)
1312adantl 480 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑓 Fn 𝑛)
14 bnj910.13 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
1514bnj941 34531 . . . . 5 (𝐶 ∈ V → ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
16153impib 1113 . . . 4 ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
179, 10, 13, 16syl3anc 1368 . . 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 34697 . . 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 34704 . . . 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 34703 . . . 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 34702 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜓″)
263, 17, 20, 25bnj951 34534 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
27 bnj910.6 . . . 4 (𝜒′[𝑝 / 𝑛]𝜒)
28 vex 3465 . . . 4 𝑝 ∈ V
291, 18, 21, 27, 28bnj919 34526 . . 3 (𝜒′ ↔ (𝑝𝐷𝑓 Fn 𝑝𝜑′𝜓′))
30 bnj910.9 . . 3 (𝜒″[𝐺 / 𝑓]𝜒′)
3114bnj918 34525 . . 3 𝐺 ∈ V
3229, 19, 22, 30, 31bnj976 34536 . 2 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
3326, 32sylibr 233 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜒″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  {cab 2702  wral 3050  wrex 3059  Vcvv 3461  [wsbc 3773  cdif 3941  cun 3942  c0 4322  {csn 4630  cop 4636   ciun 4997  suc csuc 6373   Fn wfn 6544  cfv 6549  ωcom 7871  w-bnj17 34445   predc-bnj14 34447   FrSe w-bnj15 34451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741  ax-reg 9617
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-res 5690  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-fv 6557  df-om 7872  df-bnj17 34446  df-bnj14 34448  df-bnj13 34450  df-bnj15 34452
This theorem is referenced by:  bnj998  34716
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