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Theorem bnj852 31323
Description: Technical lemma for bnj69 31410. 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 3364 . . . . . 6 (𝑋𝐴 → ∃𝑥 𝑥 = 𝑋)
21adantl 467 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥 𝑥 = 𝑋)
32ancri 531 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → (∃𝑥 𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)))
43bnj534 31140 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥(𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)))
5 eleq1 2837 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
65anbi2d 606 . . . . . 6 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑥𝐴) ↔ (𝑅 FrSe 𝐴𝑋𝐴)))
76biimpar 463 . . . . 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 8703 . . . . . . . . . 10 ω ∈ V
11 difexg 4939 . . . . . . . . . 10 (ω ∈ V → (ω ∖ {∅}) ∈ V)
1210, 11ax-mp 5 . . . . . . . . 9 (ω ∖ {∅}) ∈ V
139, 12eqeltri 2845 . . . . . . . 8 𝐷 ∈ V
14 zfregfr 8664 . . . . . . . 8 E Fr 𝐷
158, 13, 14bnj157 31261 . . . . . . 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 31282 . . . . . . . . 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 31322 . . . . . . . . 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 3025 . . . . . . . 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 397 . . . . . . 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 3074 . . . . . 6 𝑛𝐷 ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))
24 r19.21v 3108 . . . . . 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 31317 . . . . . . . . . 10 (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
2827eqeq2d 2780 . . . . . . . . 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 1551 . . . . . . 7 (𝑥 = 𝑋 → ((𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓)))
3231eubidv 2637 . . . . . 6 (𝑥 = 𝑋 → (∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3332ralbidv 3134 . . . . 5 (𝑥 = 𝑋 → (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3433adantr 466 . . . 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 31147 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
3736bnj937 31174 1 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wex 1851  wcel 2144  ∃!weu 2617  wral 3060  Vcvv 3349  [wsbc 3585  cdif 3718  c0 4061  {csn 4314   ciun 4652   class class class wbr 4784   E cep 5161  suc csuc 5868   Fn wfn 6026  cfv 6031  ωcom 7211  1𝑜c1o 7705   predc-bnj14 31088   FrSe w-bnj15 31092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-reg 8652  ax-inf2 8701
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7212  df-1o 7712  df-bnj17 31087  df-bnj14 31089  df-bnj13 31091  df-bnj15 31093
This theorem is referenced by:  bnj864  31324  bnj865  31325  bnj906  31332
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