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Theorem bnj852 31508
Description: Technical lemma for bnj69 31595. 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 3403 . . . . . 6 (𝑋𝐴 → ∃𝑥 𝑥 = 𝑋)
21adantl 474 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥 𝑥 = 𝑋)
32ancri 546 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → (∃𝑥 𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)))
43bnj534 31326 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥(𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)))
5 eleq1 2866 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
65anbi2d 623 . . . . . 6 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑥𝐴) ↔ (𝑅 FrSe 𝐴𝑋𝐴)))
76biimpar 470 . . . . 5 ((𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)) → (𝑅 FrSe 𝐴𝑥𝐴))
8 biid 253 . . . . . . . 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 8790 . . . . . . . . . 10 ω ∈ V
11 difexg 5003 . . . . . . . . . 10 (ω ∈ V → (ω ∖ {∅}) ∈ V)
1210, 11ax-mp 5 . . . . . . . . 9 (ω ∖ {∅}) ∈ V
139, 12eqeltri 2874 . . . . . . . 8 𝐷 ∈ V
14 zfregfr 8751 . . . . . . . 8 E Fr 𝐷
158, 13, 14bnj157 31446 . . . . . . 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 253 . . . . . . . . . 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 253 . . . . . . . . . 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 31467 . . . . . . . . 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 31507 . . . . . . . . 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 3054 . . . . . . . 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 402 . . . . . . 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 3107 . . . . . 6 𝑛𝐷 ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓))
24 r19.21v 3141 . . . . . 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 222 . . . . 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 31502 . . . . . . . . . 10 (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
2827eqeq2d 2809 . . . . . . . . 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 281 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ 𝜑))
31303anbi2d 1566 . . . . . . 7 (𝑥 = 𝑋 → ((𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓)))
3231eubidv 2626 . . . . . 6 (𝑥 = 𝑋 → (∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3332ralbidv 3167 . . . . 5 (𝑥 = 𝑋 → (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3433adantr 473 . . . 4 ((𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)) → (∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓) ↔ ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3526, 34mpbid 224 . . 3 ((𝑥 = 𝑋 ∧ (𝑅 FrSe 𝐴𝑋𝐴)) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
364, 35bnj593 31332 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
3736bnj937 31359 1 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑛𝐷 ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wex 1875  wcel 2157  ∃!weu 2608  wral 3089  Vcvv 3385  [wsbc 3633  cdif 3766  c0 4115  {csn 4368   ciun 4710   class class class wbr 4843   E cep 5224  suc csuc 5943   Fn wfn 6096  cfv 6101  ωcom 7299  1𝑜c1o 7792   predc-bnj14 31274   FrSe w-bnj15 31278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-reg 8739  ax-inf2 8788
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-om 7300  df-1o 7799  df-bnj17 31273  df-bnj14 31275  df-bnj13 31277  df-bnj15 31279
This theorem is referenced by:  bnj864  31509  bnj865  31510  bnj906  31517
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