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Theorem bnj852 31329
Description: Technical lemma for bnj69 31416. 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 3367 . . . . . 6 (𝑋𝐴 → ∃𝑥 𝑥 = 𝑋)
21adantl 467 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥 𝑥 = 𝑋)
32ancri 539 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → (∃𝑥 𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)))
43bnj534 31146 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥(𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)))
5 eleq1 2838 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
65anbi2d 614 . . . . . 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 8704 . . . . . . . . . 10 ω ∈ V
11 difexg 4942 . . . . . . . . . 10 (ω ∈ V → (ω ∖ {∅}) ∈ V)
1210, 11ax-mp 5 . . . . . . . . 9 (ω ∖ {∅}) ∈ V
139, 12eqeltri 2846 . . . . . . . 8 𝐷 ∈ V
14 zfregfr 8665 . . . . . . . 8 E Fr 𝐷
158, 13, 14bnj157 31267 . . . . . . 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 31288 . . . . . . . . 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 31328 . . . . . . . . 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 3026 . . . . . . . 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 3075 . . . . . 6 𝑛𝐷 ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))
24 r19.21v 3109 . . . . . 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 31323 . . . . . . . . . 10 (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
2827eqeq2d 2781 . . . . . . . . 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 1552 . . . . . . 7 (𝑥 = 𝑋 → ((𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓)))
3231eubidv 2638 . . . . . 6 (𝑥 = 𝑋 → (∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3332ralbidv 3135 . . . . 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 31153 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
3736bnj937 31180 1 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  ∃!weu 2618  wral 3061  Vcvv 3351  [wsbc 3587  cdif 3720  c0 4063  {csn 4316   ciun 4654   class class class wbr 4786   E cep 5161  suc csuc 5868   Fn wfn 6026  cfv 6031  ωcom 7212  1𝑜c1o 7706   predc-bnj14 31094   FrSe w-bnj15 31098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-reg 8653  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  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 7213  df-1o 7713  df-bnj17 31093  df-bnj14 31095  df-bnj13 31097  df-bnj15 31099
This theorem is referenced by:  bnj864  31330  bnj865  31331  bnj906  31338
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