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Theorem bnj580 31109
Description: Technical lemma for bnj579 31110. 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 1096 . . . . . 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 31104 . . . . . . 7 (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
76simp1bi 1096 . . . . . 6 (𝜒′𝑔 Fn 𝑛)
82, 7bnj240 30893 . . . . 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 31074 . . . . . . . . . . . . 13 (𝜑′ ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
13 vex 3234 . . . . . . . . . . . . . 14 𝑔 ∈ V
1410, 4, 13bnj540 31088 . . . . . . . . . . . . 13 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
15 bnj580.8 . . . . . . . . . . . . 13 (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
1615bnj591 31107 . . . . . . . . . . . . 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 31108 . . . . . . . . . . . 12 ((𝑗𝑛𝜏) → 𝜃)
1918ex 449 . . . . . . . . . . 11 (𝑗𝑛 → (𝜏𝜃))
2019rgen 2951 . . . . . . . . . 10 𝑗𝑛 (𝜏𝜃)
21 vex 3234 . . . . . . . . . . 11 𝑛 ∈ V
2221, 17bnj110 31054 . . . . . . . . . 10 (( E Fr 𝑛 ∧ ∀𝑗𝑛 (𝜏𝜃)) → ∀𝑗𝑛 𝜃)
2320, 22mpan2 707 . . . . . . . . 9 ( E Fr 𝑛 → ∀𝑗𝑛 𝜃)
2415ralbii 3009 . . . . . . . . 9 (∀𝑗𝑛 𝜃 ↔ ∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2523, 24sylib 208 . . . . . . . 8 ( E Fr 𝑛 → ∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2625r19.21be 2962 . . . . . . 7 𝑗𝑛 ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2711bnj923 30964 . . . . . . . . . . . . 13 (𝑛𝐷𝑛 ∈ ω)
28 nnord 7115 . . . . . . . . . . . . 13 (𝑛 ∈ ω → Ord 𝑛)
29 ordfr 5776 . . . . . . . . . . . . 13 (Ord 𝑛 → E Fr 𝑛)
3027, 28, 293syl 18 . . . . . . . . . . . 12 (𝑛𝐷 → E Fr 𝑛)
31303ad2ant1 1102 . . . . . . . . . . 11 ((𝑛𝐷𝜒𝜒′) → E Fr 𝑛)
3231pm4.71ri 666 . . . . . . . . . 10 ((𝑛𝐷𝜒𝜒′) ↔ ( E Fr 𝑛 ∧ (𝑛𝐷𝜒𝜒′)))
3332imbi1i 338 . . . . . . . . 9 (((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ (( E Fr 𝑛 ∧ (𝑛𝐷𝜒𝜒′)) → (𝑓𝑗) = (𝑔𝑗)))
34 impexp 461 . . . . . . . . 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 3009 . . . . . . 7 (∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ∀𝑗𝑛 ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))))
3726, 36mpbir 221 . . . . . 6 𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))
38 r19.21v 2989 . . . . . 6 (∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷𝜒𝜒′) → ∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗)))
3937, 38mpbi 220 . . . . 5 ((𝑛𝐷𝜒𝜒′) → ∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗))
40 eqfnfv 6351 . . . . . 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 1287 . . 3 (𝑛𝐷 → ((𝜒𝜒′) → 𝑓 = 𝑔))
4443alrimivv 1896 . 2 (𝑛𝐷 → ∀𝑓𝑔((𝜒𝜒′) → 𝑓 = 𝑔))
45 sbsbc 3472 . . . . . 6 ([𝑔 / 𝑓]𝜒[𝑔 / 𝑓]𝜒)
4645anbi2i 730 . . . . 5 ((𝜒 ∧ [𝑔 / 𝑓]𝜒) ↔ (𝜒[𝑔 / 𝑓]𝜒))
4746imbi1i 338 . . . 4 (((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔) ↔ ((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
48472albii 1788 . . 3 (∀𝑓𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔) ↔ ∀𝑓𝑔((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
49 nfv 1883 . . . 4 𝑔𝜒
5049mo3 2536 . . 3 (∃*𝑓𝜒 ↔ ∀𝑓𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
515anbi2i 730 . . . . 5 ((𝜒𝜒′) ↔ (𝜒[𝑔 / 𝑓]𝜒))
5251imbi1i 338 . . . 4 (((𝜒𝜒′) → 𝑓 = 𝑔) ↔ ((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
53522albii 1788 . . 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 383  w3a 1054  wal 1521   = wceq 1523  [wsb 1937  wcel 2030  ∃*wmo 2499  wral 2941  [wsbc 3468  cdif 3604  c0 3948  {csn 4210   ciun 4552   class class class wbr 4685   E cep 5057   Fr wfr 5099  Ord word 5760  suc csuc 5763   Fn wfn 5921  cfv 5926  ωcom 7107   predc-bnj14 30882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-om 7108  df-bnj17 30881
This theorem is referenced by:  bnj579  31110
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