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Theorem bnj580 31861
Description: Technical lemma for bnj579 31862. 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 1125 . . . . . 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 31856 . . . . . . 7 (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
76simp1bi 1125 . . . . . 6 (𝜒′𝑔 Fn 𝑛)
82, 7bnj240 31646 . . . . 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 31826 . . . . . . . . . . . . 13 (𝜑′ ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
13 vex 3412 . . . . . . . . . . . . . 14 𝑔 ∈ V
1410, 4, 13bnj540 31840 . . . . . . . . . . . . 13 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
15 bnj580.8 . . . . . . . . . . . . 13 (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
1615bnj591 31859 . . . . . . . . . . . . 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 31860 . . . . . . . . . . . 12 ((𝑗𝑛𝜏) → 𝜃)
1918ex 405 . . . . . . . . . . 11 (𝑗𝑛 → (𝜏𝜃))
2019rgen 3092 . . . . . . . . . 10 𝑗𝑛 (𝜏𝜃)
21 vex 3412 . . . . . . . . . . 11 𝑛 ∈ V
2221, 17bnj110 31806 . . . . . . . . . 10 (( E Fr 𝑛 ∧ ∀𝑗𝑛 (𝜏𝜃)) → ∀𝑗𝑛 𝜃)
2320, 22mpan2 678 . . . . . . . . 9 ( E Fr 𝑛 → ∀𝑗𝑛 𝜃)
2415ralbii 3109 . . . . . . . . 9 (∀𝑗𝑛 𝜃 ↔ ∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2523, 24sylib 210 . . . . . . . 8 ( E Fr 𝑛 → ∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2625r19.21be 3154 . . . . . . 7 𝑗𝑛 ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2711bnj923 31716 . . . . . . . . . . . . 13 (𝑛𝐷𝑛 ∈ ω)
28 nnord 7402 . . . . . . . . . . . . 13 (𝑛 ∈ ω → Ord 𝑛)
29 ordfr 6041 . . . . . . . . . . . . 13 (Ord 𝑛 → E Fr 𝑛)
3027, 28, 293syl 18 . . . . . . . . . . . 12 (𝑛𝐷 → E Fr 𝑛)
31303ad2ant1 1113 . . . . . . . . . . 11 ((𝑛𝐷𝜒𝜒′) → E Fr 𝑛)
3231pm4.71ri 553 . . . . . . . . . 10 ((𝑛𝐷𝜒𝜒′) ↔ ( E Fr 𝑛 ∧ (𝑛𝐷𝜒𝜒′)))
3332imbi1i 342 . . . . . . . . 9 (((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ (( E Fr 𝑛 ∧ (𝑛𝐷𝜒𝜒′)) → (𝑓𝑗) = (𝑔𝑗)))
34 impexp 443 . . . . . . . . 9 ((( E Fr 𝑛 ∧ (𝑛𝐷𝜒𝜒′)) → (𝑓𝑗) = (𝑔𝑗)) ↔ ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))))
3533, 34bitri 267 . . . . . . . 8 (((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))))
3635ralbii 3109 . . . . . . 7 (∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ∀𝑗𝑛 ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))))
3726, 36mpbir 223 . . . . . 6 𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))
38 r19.21v 3119 . . . . . 6 (∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷𝜒𝜒′) → ∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗)))
3937, 38mpbi 222 . . . . 5 ((𝑛𝐷𝜒𝜒′) → ∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗))
40 eqfnfv 6625 . . . . . 6 ((𝑓 Fn 𝑛𝑔 Fn 𝑛) → (𝑓 = 𝑔 ↔ ∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗)))
4140biimprd 240 . . . . 5 ((𝑓 Fn 𝑛𝑔 Fn 𝑛) → (∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗) → 𝑓 = 𝑔))
428, 39, 41sylc 65 . . . 4 ((𝑛𝐷𝜒𝜒′) → 𝑓 = 𝑔)
43423expib 1102 . . 3 (𝑛𝐷 → ((𝜒𝜒′) → 𝑓 = 𝑔))
4443alrimivv 1887 . 2 (𝑛𝐷 → ∀𝑓𝑔((𝜒𝜒′) → 𝑓 = 𝑔))
45 sbsbc 3679 . . . . . 6 ([𝑔 / 𝑓]𝜒[𝑔 / 𝑓]𝜒)
4645anbi2i 613 . . . . 5 ((𝜒 ∧ [𝑔 / 𝑓]𝜒) ↔ (𝜒[𝑔 / 𝑓]𝜒))
4746imbi1i 342 . . . 4 (((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔) ↔ ((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
48472albii 1783 . . 3 (∀𝑓𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔) ↔ ∀𝑓𝑔((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
49 nfv 1873 . . . 4 𝑔𝜒
5049mo3 2577 . . 3 (∃*𝑓𝜒 ↔ ∀𝑓𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
515anbi2i 613 . . . . 5 ((𝜒𝜒′) ↔ (𝜒[𝑔 / 𝑓]𝜒))
5251imbi1i 342 . . . 4 (((𝜒𝜒′) → 𝑓 = 𝑔) ↔ ((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
53522albii 1783 . . 3 (∀𝑓𝑔((𝜒𝜒′) → 𝑓 = 𝑔) ↔ ∀𝑓𝑔((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
5448, 50, 533bitr4i 295 . 2 (∃*𝑓𝜒 ↔ ∀𝑓𝑔((𝜒𝜒′) → 𝑓 = 𝑔))
5544, 54sylibr 226 1 (𝑛𝐷 → ∃*𝑓𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068  wal 1505   = wceq 1507  [wsb 2015  wcel 2050  ∃*wmo 2545  wral 3082  [wsbc 3675  cdif 3820  c0 4172  {csn 4435   ciun 4788   class class class wbr 4925   E cep 5312   Fr wfr 5359  Ord word 6025  suc csuc 6028   Fn wfn 6180  cfv 6185  ωcom 7394   predc-bnj14 31635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-fv 6193  df-om 7395  df-bnj17 31634
This theorem is referenced by:  bnj579  31862
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