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Theorem bnj580 32293
 Description: Technical lemma for bnj579 32294. 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 1142 . . . . . 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 32288 . . . . . . 7 (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
76simp1bi 1142 . . . . . 6 (𝜒′𝑔 Fn 𝑛)
82, 7bnj240 32077 . . . . 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 32258 . . . . . . . . . . . . 13 (𝜑′ ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
13 vex 3447 . . . . . . . . . . . . . 14 𝑔 ∈ V
1410, 4, 13bnj540 32272 . . . . . . . . . . . . 13 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
15 bnj580.8 . . . . . . . . . . . . 13 (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
1615bnj591 32291 . . . . . . . . . . . . 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 32292 . . . . . . . . . . . 12 ((𝑗𝑛𝜏) → 𝜃)
1918ex 416 . . . . . . . . . . 11 (𝑗𝑛 → (𝜏𝜃))
2019rgen 3119 . . . . . . . . . 10 𝑗𝑛 (𝜏𝜃)
21 vex 3447 . . . . . . . . . . 11 𝑛 ∈ V
2221, 17bnj110 32238 . . . . . . . . . 10 (( E Fr 𝑛 ∧ ∀𝑗𝑛 (𝜏𝜃)) → ∀𝑗𝑛 𝜃)
2320, 22mpan2 690 . . . . . . . . 9 ( E Fr 𝑛 → ∀𝑗𝑛 𝜃)
2415ralbii 3136 . . . . . . . . 9 (∀𝑗𝑛 𝜃 ↔ ∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2523, 24sylib 221 . . . . . . . 8 ( E Fr 𝑛 → ∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2625r19.21be 3177 . . . . . . 7 𝑗𝑛 ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2711bnj923 32147 . . . . . . . . . . . . 13 (𝑛𝐷𝑛 ∈ ω)
28 nnord 7572 . . . . . . . . . . . . 13 (𝑛 ∈ ω → Ord 𝑛)
29 ordfr 6178 . . . . . . . . . . . . 13 (Ord 𝑛 → E Fr 𝑛)
3027, 28, 293syl 18 . . . . . . . . . . . 12 (𝑛𝐷 → E Fr 𝑛)
31303ad2ant1 1130 . . . . . . . . . . 11 ((𝑛𝐷𝜒𝜒′) → E Fr 𝑛)
3231pm4.71ri 564 . . . . . . . . . 10 ((𝑛𝐷𝜒𝜒′) ↔ ( E Fr 𝑛 ∧ (𝑛𝐷𝜒𝜒′)))
3332imbi1i 353 . . . . . . . . 9 (((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ (( E Fr 𝑛 ∧ (𝑛𝐷𝜒𝜒′)) → (𝑓𝑗) = (𝑔𝑗)))
34 impexp 454 . . . . . . . . 9 ((( E Fr 𝑛 ∧ (𝑛𝐷𝜒𝜒′)) → (𝑓𝑗) = (𝑔𝑗)) ↔ ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))))
3533, 34bitri 278 . . . . . . . 8 (((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))))
3635ralbii 3136 . . . . . . 7 (∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ∀𝑗𝑛 ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))))
3726, 36mpbir 234 . . . . . 6 𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))
38 r19.21v 3145 . . . . . 6 (∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷𝜒𝜒′) → ∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗)))
3937, 38mpbi 233 . . . . 5 ((𝑛𝐷𝜒𝜒′) → ∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗))
40 eqfnfv 6783 . . . . . 6 ((𝑓 Fn 𝑛𝑔 Fn 𝑛) → (𝑓 = 𝑔 ↔ ∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗)))
4140biimprd 251 . . . . 5 ((𝑓 Fn 𝑛𝑔 Fn 𝑛) → (∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗) → 𝑓 = 𝑔))
428, 39, 41sylc 65 . . . 4 ((𝑛𝐷𝜒𝜒′) → 𝑓 = 𝑔)
43423expib 1119 . . 3 (𝑛𝐷 → ((𝜒𝜒′) → 𝑓 = 𝑔))
4443alrimivv 1929 . 2 (𝑛𝐷 → ∀𝑓𝑔((𝜒𝜒′) → 𝑓 = 𝑔))
45 sbsbc 3727 . . . . . 6 ([𝑔 / 𝑓]𝜒[𝑔 / 𝑓]𝜒)
4645anbi2i 625 . . . . 5 ((𝜒 ∧ [𝑔 / 𝑓]𝜒) ↔ (𝜒[𝑔 / 𝑓]𝜒))
4746imbi1i 353 . . . 4 (((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔) ↔ ((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
48472albii 1822 . . 3 (∀𝑓𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔) ↔ ∀𝑓𝑔((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
49 nfv 1915 . . . 4 𝑔𝜒
5049mo3 2626 . . 3 (∃*𝑓𝜒 ↔ ∀𝑓𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
515anbi2i 625 . . . . 5 ((𝜒𝜒′) ↔ (𝜒[𝑔 / 𝑓]𝜒))
5251imbi1i 353 . . . 4 (((𝜒𝜒′) → 𝑓 = 𝑔) ↔ ((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
53522albii 1822 . . 3 (∀𝑓𝑔((𝜒𝜒′) → 𝑓 = 𝑔) ↔ ∀𝑓𝑔((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
5448, 50, 533bitr4i 306 . 2 (∃*𝑓𝜒 ↔ ∀𝑓𝑔((𝜒𝜒′) → 𝑓 = 𝑔))
5544, 54sylibr 237 1 (𝑛𝐷 → ∃*𝑓𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538  [wsb 2069   ∈ wcel 2112  ∃*wmo 2599  ∀wral 3109  [wsbc 3723   ∖ cdif 3881  ∅c0 4246  {csn 4528  ∪ ciun 4884   class class class wbr 5033   E cep 5432   Fr wfr 5479  Ord word 6162  suc csuc 6165   Fn wfn 6323  ‘cfv 6328  ωcom 7564   predc-bnj14 32066 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-om 7565  df-bnj17 32065 This theorem is referenced by:  bnj579  32294
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