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Theorem bnj580 31321
Description: Technical lemma for bnj579 31322. 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 1139 . . . . . 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 31316 . . . . . . 7 (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
76simp1bi 1139 . . . . . 6 (𝜒′𝑔 Fn 𝑛)
82, 7bnj240 31105 . . . . 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 31286 . . . . . . . . . . . . 13 (𝜑′ ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
13 vex 3354 . . . . . . . . . . . . . 14 𝑔 ∈ V
1410, 4, 13bnj540 31300 . . . . . . . . . . . . 13 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
15 bnj580.8 . . . . . . . . . . . . 13 (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
1615bnj591 31319 . . . . . . . . . . . . 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 31320 . . . . . . . . . . . 12 ((𝑗𝑛𝜏) → 𝜃)
1918ex 397 . . . . . . . . . . 11 (𝑗𝑛 → (𝜏𝜃))
2019rgen 3071 . . . . . . . . . 10 𝑗𝑛 (𝜏𝜃)
21 vex 3354 . . . . . . . . . . 11 𝑛 ∈ V
2221, 17bnj110 31266 . . . . . . . . . 10 (( E Fr 𝑛 ∧ ∀𝑗𝑛 (𝜏𝜃)) → ∀𝑗𝑛 𝜃)
2320, 22mpan2 671 . . . . . . . . 9 ( E Fr 𝑛 → ∀𝑗𝑛 𝜃)
2415ralbii 3129 . . . . . . . . 9 (∀𝑗𝑛 𝜃 ↔ ∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2523, 24sylib 208 . . . . . . . 8 ( E Fr 𝑛 → ∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2625r19.21be 3082 . . . . . . 7 𝑗𝑛 ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
2711bnj923 31176 . . . . . . . . . . . . 13 (𝑛𝐷𝑛 ∈ ω)
28 nnord 7220 . . . . . . . . . . . . 13 (𝑛 ∈ ω → Ord 𝑛)
29 ordfr 5881 . . . . . . . . . . . . 13 (Ord 𝑛 → E Fr 𝑛)
3027, 28, 293syl 18 . . . . . . . . . . . 12 (𝑛𝐷 → E Fr 𝑛)
31303ad2ant1 1127 . . . . . . . . . . 11 ((𝑛𝐷𝜒𝜒′) → E Fr 𝑛)
3231pm4.71ri 550 . . . . . . . . . 10 ((𝑛𝐷𝜒𝜒′) ↔ ( E Fr 𝑛 ∧ (𝑛𝐷𝜒𝜒′)))
3332imbi1i 338 . . . . . . . . 9 (((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ (( E Fr 𝑛 ∧ (𝑛𝐷𝜒𝜒′)) → (𝑓𝑗) = (𝑔𝑗)))
34 impexp 437 . . . . . . . . 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 3129 . . . . . . 7 (∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ∀𝑗𝑛 ( E Fr 𝑛 → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))))
3726, 36mpbir 221 . . . . . 6 𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗))
38 r19.21v 3109 . . . . . 6 (∀𝑗𝑛 ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷𝜒𝜒′) → ∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗)))
3937, 38mpbi 220 . . . . 5 ((𝑛𝐷𝜒𝜒′) → ∀𝑗𝑛 (𝑓𝑗) = (𝑔𝑗))
40 eqfnfv 6454 . . . . . 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 1116 . . 3 (𝑛𝐷 → ((𝜒𝜒′) → 𝑓 = 𝑔))
4443alrimivv 2008 . 2 (𝑛𝐷 → ∀𝑓𝑔((𝜒𝜒′) → 𝑓 = 𝑔))
45 sbsbc 3591 . . . . . 6 ([𝑔 / 𝑓]𝜒[𝑔 / 𝑓]𝜒)
4645anbi2i 609 . . . . 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 1896 . . 3 (∀𝑓𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔) ↔ ∀𝑓𝑔((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
49 nfv 1995 . . . 4 𝑔𝜒
5049mo3 2656 . . 3 (∃*𝑓𝜒 ↔ ∀𝑓𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
515anbi2i 609 . . . . 5 ((𝜒𝜒′) ↔ (𝜒[𝑔 / 𝑓]𝜒))
5251imbi1i 338 . . . 4 (((𝜒𝜒′) → 𝑓 = 𝑔) ↔ ((𝜒[𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
53522albii 1896 . . 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 382  w3a 1071  wal 1629   = wceq 1631  [wsb 2049  wcel 2145  ∃*wmo 2619  wral 3061  [wsbc 3587  cdif 3720  c0 4063  {csn 4316   ciun 4654   class class class wbr 4786   E cep 5161   Fr wfr 5205  Ord word 5865  suc csuc 5868   Fn wfn 6026  cfv 6031  ωcom 7212   predc-bnj14 31094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-om 7213  df-bnj17 31093
This theorem is referenced by:  bnj579  31322
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