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Theorem bnj852 30734
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
bnj852.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj852.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj852.3 𝐷 = (ω ∖ {∅})
Assertion
Ref Expression
bnj852 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑓,𝑖,𝑛   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑛
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑛)   𝐷(𝑦)   𝑋(𝑦,𝑖)

Proof of Theorem bnj852
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elisset 3204 . . . . . 6 (𝑋𝐴 → ∃𝑥 𝑥 = 𝑋)
21adantl 482 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥 𝑥 = 𝑋)
32ancri 574 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → (∃𝑥 𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)))
43bnj534 30551 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥(𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)))
5 eleq1 2686 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
65anbi2d 739 . . . . . 6 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑥𝐴) ↔ (𝑅 FrSe 𝐴𝑋𝐴)))
76biimpar 502 . . . . 5 ((𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)) → (𝑅 FrSe 𝐴𝑥𝐴))
8 biid 251 . . . . . . . 8 (∀𝑧𝐷 (𝑧 E 𝑛[𝑧 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))) ↔ ∀𝑧𝐷 (𝑧 E 𝑛[𝑧 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))))
9 bnj852.3 . . . . . . . . 9 𝐷 = (ω ∖ {∅})
10 omex 8492 . . . . . . . . . 10 ω ∈ V
11 difexg 4773 . . . . . . . . . 10 (ω ∈ V → (ω ∖ {∅}) ∈ V)
1210, 11ax-mp 5 . . . . . . . . 9 (ω ∖ {∅}) ∈ V
139, 12eqeltri 2694 . . . . . . . 8 𝐷 ∈ V
14 zfregfr 8461 . . . . . . . 8 E Fr 𝐷
158, 13, 14bnj157 30672 . . . . . . 7 (∀𝑛𝐷 (∀𝑧𝐷 (𝑧 E 𝑛[𝑧 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))) → ∀𝑛𝐷 ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))
16 biid 251 . . . . . . . . . 10 ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
17 bnj852.2 . . . . . . . . . 10 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
18 biid 251 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))
1916, 17, 9, 18, 8bnj153 30693 . . . . . . . . 9 (𝑛 = 1𝑜 → ((𝑛𝐷 ∧ ∀𝑧𝐷 (𝑧 E 𝑛[𝑧 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))))
2016, 17, 9, 18, 8bnj601 30733 . . . . . . . . 9 (𝑛 ≠ 1𝑜 → ((𝑛𝐷 ∧ ∀𝑧𝐷 (𝑧 E 𝑛[𝑧 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))))
2119, 20pm2.61ine 2873 . . . . . . . 8 ((𝑛𝐷 ∧ ∀𝑧𝐷 (𝑧 E 𝑛[𝑧 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))
2221ex 450 . . . . . . 7 (𝑛𝐷 → (∀𝑧𝐷 (𝑧 E 𝑛[𝑧 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))))
2315, 22mprg 2921 . . . . . 6 𝑛𝐷 ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))
24 r19.21v 2955 . . . . . 6 (∀𝑛𝐷 ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓)))
2523, 24mpbi 220 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))
267, 25syl 17 . . . 4 ((𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))
27 bnj602 30728 . . . . . . . . . 10 (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
2827eqeq2d 2631 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
29 bnj852.1 . . . . . . . . 9 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
3028, 29syl6bbr 278 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ 𝜑))
31303anbi2d 1401 . . . . . . 7 (𝑥 = 𝑋 → ((𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓)))
3231eubidv 2489 . . . . . 6 (𝑥 = 𝑋 → (∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3332ralbidv 2981 . . . . 5 (𝑥 = 𝑋 → (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3433adantr 481 . . . 4 ((𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)) → (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3526, 34mpbid 222 . . 3 ((𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
364, 35bnj593 30558 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
3736bnj937 30585 1 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  ∃!weu 2469  wral 2907  Vcvv 3189  [wsbc 3421  cdif 3556  c0 3896  {csn 4153   ciun 4490   class class class wbr 4618   E cep 4988  suc csuc 5689   Fn wfn 5847  cfv 5852  ωcom 7019  1𝑜c1o 7505   predc-bnj14 30496   FrSe w-bnj15 30500
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-pow 4808  ax-pr 4872  ax-un 6909  ax-reg 8449  ax-inf2 8490
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  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-1o 7512  df-bnj17 30495  df-bnj14 30497  df-bnj13 30499  df-bnj15 30501
This theorem is referenced by:  bnj864  30735  bnj865  30736  bnj906  30743
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