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Theorem bnj998 30769
Description: Technical lemma for bnj69 30821. 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
bnj998.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj998.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj998.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj998.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj998.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj998.7 (𝜑′[𝑝 / 𝑛]𝜑)
bnj998.8 (𝜓′[𝑝 / 𝑛]𝜓)
bnj998.9 (𝜒′[𝑝 / 𝑛]𝜒)
bnj998.10 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj998.11 (𝜓″[𝐺 / 𝑓]𝜓′)
bnj998.12 (𝜒″[𝐺 / 𝑓]𝜒′)
bnj998.13 𝐷 = (ω ∖ {∅})
bnj998.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj998.15 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj998.16 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
Assertion
Ref Expression
bnj998 ((𝜃𝜒𝜏𝜂) → 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,𝑧,𝑓,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜃(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜏(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑧,𝑝)   𝐵(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑚,𝑝)   𝑅(𝑧,𝑝)   𝐺(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑧,𝑚,𝑝)   𝜑′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj998
StepHypRef Expression
1 bnj998.4 . . . . . 6 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
2 bnj253 30512 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
32simp1bi 1074 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → (𝑅 FrSe 𝐴𝑋𝐴))
41, 3sylbi 207 . . . . 5 (𝜃 → (𝑅 FrSe 𝐴𝑋𝐴))
54bnj705 30566 . . . 4 ((𝜃𝜒𝜏𝜂) → (𝑅 FrSe 𝐴𝑋𝐴))
6 bnj643 30562 . . . 4 ((𝜃𝜒𝜏𝜂) → 𝜒)
7 bnj998.5 . . . . . 6 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
8 3simpc 1058 . . . . . 6 ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
97, 8sylbi 207 . . . . 5 (𝜏 → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
109bnj707 30568 . . . 4 ((𝜃𝜒𝜏𝜂) → (𝑛 = suc 𝑚𝑝 = suc 𝑛))
11 bnj255 30513 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜒 ∧ (𝑛 = suc 𝑚𝑝 = suc 𝑛)))
125, 6, 10, 11syl3anbrc 1244 . . 3 ((𝜃𝜒𝜏𝜂) → ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛))
13 bnj252 30511 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)))
1412, 13sylib 208 . 2 ((𝜃𝜒𝜏𝜂) → ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)))
15 bnj998.1 . . 3 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
16 bnj998.2 . . 3 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
17 bnj998.3 . . 3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
18 bnj998.7 . . 3 (𝜑′[𝑝 / 𝑛]𝜑)
19 bnj998.8 . . 3 (𝜓′[𝑝 / 𝑛]𝜓)
20 bnj998.9 . . 3 (𝜒′[𝑝 / 𝑛]𝜒)
21 bnj998.10 . . 3 (𝜑″[𝐺 / 𝑓]𝜑′)
22 bnj998.11 . . 3 (𝜓″[𝐺 / 𝑓]𝜓′)
23 bnj998.12 . . 3 (𝜒″[𝐺 / 𝑓]𝜒′)
24 bnj998.13 . . 3 𝐷 = (ω ∖ {∅})
25 bnj998.14 . . 3 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
26 bnj998.15 . . 3 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
27 bnj998.16 . . 3 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
28 biid 251 . . 3 ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓))
29 biid 251 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑚𝑛) ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
3015, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29bnj910 30761 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜒″)
3114, 30syl 17 1 ((𝜃𝜒𝜏𝜂) → 𝜒″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  [wsbc 3421  cdif 3556  cun 3557  c0 3896  {csn 4153  cop 4159   ciun 4490  suc csuc 5689   Fn wfn 5847  cfv 5852  ωcom 7019  w-bnj17 30494   predc-bnj14 30496   FrSe w-bnj15 30500   trClc-bnj18 30502
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 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909  ax-reg 8449
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  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 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-bnj17 30495  df-bnj14 30497  df-bnj13 30499  df-bnj15 30501
This theorem is referenced by:  bnj1020  30776
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