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Theorem bnj580 30718
Description: Technical lemma for bnj579 30719. 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
bnj580.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj580.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj580.3 (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))
bnj580.4 (𝜑′[𝑔 / 𝑓]𝜑)
bnj580.5 (𝜓′[𝑔 / 𝑓]𝜓)
bnj580.6 (𝜒′[𝑔 / 𝑓]𝜒)
bnj580.7 𝐷 = (ω ∖ {∅})
bnj580.8 (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
bnj580.9 (𝜏 ↔ ∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]𝜃))
Assertion
Ref Expression
bnj580 (𝑛𝐷 → ∃*𝑓𝜒)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑘   𝐷,𝑓,𝑔,𝑗,𝑘   𝑅,𝑓,𝑖,𝑘   𝜒,𝑔,𝑗,𝑘   𝑗,𝜒′,𝑘   𝑓,𝑛   𝑔,𝑖,𝑛,𝑘   𝑥,𝑓   𝑦,𝑓,𝑔,𝑖,𝑘   𝑗,𝑛   𝜃,𝑘
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝜒(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜃(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑛)   𝜏(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝐴(𝑥,𝑦,𝑔,𝑗,𝑛)   𝐷(𝑥,𝑦,𝑖,𝑛)   𝑅(𝑥,𝑦,𝑔,𝑗,𝑛)   𝜑′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝜒′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑛)

Proof of Theorem bnj580
StepHypRef Expression
1 bnj580.3 . . . . . . 7 (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))
21simp1bi 1074 . . . . . 6 (𝜒𝑓 Fn 𝑛)
3 bnj580.4 . . . . . . . 8 (𝜑′[𝑔 / 𝑓]𝜑)
4 bnj580.5 . . . . . . . 8 (𝜓′[𝑔 / 𝑓]𝜓)
5 bnj580.6 . . . . . . . 8 (𝜒′[𝑔 / 𝑓]𝜒)
61, 3, 4, 5bnj581 30713 . . . . . . 7 (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
76simp1bi 1074 . . . . . 6 (𝜒′𝑔 Fn 𝑛)
82, 7bnj240 30499 . . . . 5 ((𝑛𝐷𝜒𝜒′) → (𝑓 Fn 𝑛𝑔 Fn 𝑛))
9 bnj580.1 . . . . . . . . . . . . 13 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
10 bnj580.2 . . . . . . . . . . . . 13 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
11 bnj580.7 . . . . . . . . . . . . 13 𝐷 = (ω ∖ {∅})
123, 9bnj154 30683 . . . . . . . . . . . . 13 (𝜑′ ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
13 vex 3192 . . . . . . . . . . . . . 14 𝑔 ∈ V
1410, 4, 13bnj540 30697 . . . . . . . . . . . . 13 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
15 bnj580.8 . . . . . . . . . . . . 13 (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
1615bnj591 30716 . . . . . . . . . . . . 13 ([𝑘 / 𝑗]𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))
17 bnj580.9 . . . . . . . . . . . . 13 (𝜏 ↔ ∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]𝜃))
189, 10, 1, 11, 12, 14, 6, 15, 16, 17bnj594 30717 . . . . . . . . . . . 12 ((𝑗𝑛𝜏) → 𝜃)
1918ex 450 . . . . . . . . . . 11 (𝑗𝑛 → (𝜏𝜃))
2019rgen 2917 . . . . . . . . . 10 𝑗𝑛 (𝜏𝜃)
21 vex 3192 . . . . . . . . . . 11 𝑛 ∈ V
2221, 17bnj110 30663 . . . . . . . . . 10 (( E Fr 𝑛 ∧ ∀𝑗𝑛 (𝜏𝜃)) → ∀𝑗𝑛 𝜃)
2320, 22mpan2 706 . . . . . . . . 9 ( E Fr 𝑛 → ∀𝑗𝑛 𝜃)
2415ralbii 2975 . . . . . . . . 9 (∀𝑗𝑛 𝜃 ↔ ∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2523, 24sylib 208 . . . . . . . 8 ( E Fr 𝑛 → ∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2625r19.21be 2928 . . . . . . 7 𝑗𝑛 ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2711bnj923 30573 . . . . . . . . . . . . 13 (𝑛𝐷𝑛 ∈ ω)
28 nnord 7027 . . . . . . . . . . . . 13 (𝑛 ∈ ω → Ord 𝑛)
29 ordfr 5702 . . . . . . . . . . . . 13 (Ord 𝑛 → E Fr 𝑛)
3027, 28, 293syl 18 . . . . . . . . . . . 12 (𝑛𝐷 → E Fr 𝑛)
31303ad2ant1 1080 . . . . . . . . . . 11 ((𝑛𝐷𝜒𝜒′) → E Fr 𝑛)
3231pm4.71ri 664 . . . . . . . . . 10 ((𝑛𝐷𝜒𝜒′) ↔ ( E Fr 𝑛 ∧ (𝑛𝐷𝜒𝜒′)))
3332imbi1i 339 . . . . . . . . 9 (((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ (( E Fr 𝑛 ∧ (𝑛𝐷𝜒𝜒′)) → (𝑓𝑗) = (𝑔𝑗)))
34 impexp 462 . . . . . . . . 9 ((( E Fr 𝑛 ∧ (𝑛𝐷𝜒𝜒′)) → (𝑓𝑗) = (𝑔𝑗)) ↔ ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))))
3533, 34bitri 264 . . . . . . . 8 (((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))))
3635ralbii 2975 . . . . . . 7 (∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ∀𝑗𝑛 ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))))
3726, 36mpbir 221 . . . . . 6 𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))
38 r19.21v 2955 . . . . . 6 (∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷𝜒𝜒′) → ∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗)))
3937, 38mpbi 220 . . . . 5 ((𝑛𝐷𝜒𝜒′) → ∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗))
40 eqfnfv 6272 . . . . . 6 ((𝑓 Fn 𝑛𝑔 Fn 𝑛) → (𝑓 = 𝑔 ↔ ∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗)))
4140biimprd 238 . . . . 5 ((𝑓 Fn 𝑛𝑔 Fn 𝑛) → (∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗) → 𝑓 = 𝑔))
428, 39, 41sylc 65 . . . 4 ((𝑛𝐷𝜒𝜒′) → 𝑓 = 𝑔)
43423expib 1265 . . 3 (𝑛𝐷 → ((𝜒𝜒′) → 𝑓 = 𝑔))
4443alrimivv 1853 . 2 (𝑛𝐷 → ∀𝑓𝑔((𝜒𝜒′) → 𝑓 = 𝑔))
45 sbsbc 3425 . . . . . 6 ([𝑔 / 𝑓]𝜒[𝑔 / 𝑓]𝜒)
4645anbi2i 729 . . . . 5 ((𝜒 ∧ [𝑔 / 𝑓]𝜒) ↔ (𝜒[𝑔 / 𝑓]𝜒))
4746imbi1i 339 . . . 4 (((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔) ↔ ((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
48472albii 1745 . . 3 (∀𝑓𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔) ↔ ∀𝑓𝑔((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
49 nfv 1840 . . . 4 𝑔𝜒
5049mo3 2506 . . 3 (∃*𝑓𝜒 ↔ ∀𝑓𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
515anbi2i 729 . . . . 5 ((𝜒𝜒′) ↔ (𝜒[𝑔 / 𝑓]𝜒))
5251imbi1i 339 . . . 4 (((𝜒𝜒′) → 𝑓 = 𝑔) ↔ ((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
53522albii 1745 . . 3 (∀𝑓𝑔((𝜒𝜒′) → 𝑓 = 𝑔) ↔ ∀𝑓𝑔((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
5448, 50, 533bitr4i 292 . 2 (∃*𝑓𝜒 ↔ ∀𝑓𝑔((𝜒𝜒′) → 𝑓 = 𝑔))
5544, 54sylibr 224 1 (𝑛𝐷 → ∃*𝑓𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  [wsb 1877  wcel 1987  ∃*wmo 2470  wral 2907  [wsbc 3421  cdif 3556  c0 3896  {csn 4153   ciun 4490   class class class wbr 4618   E cep 4988   Fr wfr 5035  Ord word 5686  suc csuc 5689   Fn wfn 5847  cfv 5852  ωcom 7019   predc-bnj14 30488
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-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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-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-fv 5860  df-om 7020  df-bnj17 30487
This theorem is referenced by:  bnj579  30719
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