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Theorem bnj908 31333
Description: Technical lemma for bnj69 31410. 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
bnj908.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj908.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj908.3 𝐷 = (ω ∖ {∅})
bnj908.4 (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
bnj908.5 (𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))
bnj908.10 (𝜑′[𝑚 / 𝑛]𝜑)
bnj908.11 (𝜓′[𝑚 / 𝑛]𝜓)
bnj908.12 (𝜒′[𝑚 / 𝑛]𝜒)
bnj908.13 (𝜑″[𝐺 / 𝑓]𝜑)
bnj908.14 (𝜓″[𝐺 / 𝑓]𝜓)
bnj908.15 (𝜒″[𝐺 / 𝑓]𝜒)
bnj908.16 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj908.17 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj908.18 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj908.19 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj908.20 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
bnj908.21 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
bnj908.22 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
bnj908.23 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
bnj908.24 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
bnj908.25 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
bnj908.26 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
Assertion
Ref Expression
bnj908 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓(𝐺 Fn 𝑛𝜑″𝜓″))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑝   𝑦,𝐴,𝑓,𝑖,𝑛,𝑝   𝐷,𝑝   𝑖,𝐺,𝑦   𝑅,𝑓,𝑖,𝑚,𝑛,𝑝   𝑦,𝑅   𝜂,𝑓,𝑖   𝑥,𝑓,𝑚,𝑛,𝑝   𝑖,𝜑′,𝑝   𝜑,𝑚,𝑝   𝜓,𝑚,𝑝   𝜃,𝑝
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜒(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜃(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜏(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑥,𝑦,𝑚,𝑛,𝑝)   𝜁(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜌(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑥)   𝐵(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝑅(𝑥)   𝐺(𝑥,𝑓,𝑚,𝑛,𝑝)   𝐾(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐿(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑′(𝑥,𝑦,𝑓,𝑚,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒″(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj908
StepHypRef Expression
1 bnj248 31100 . . . . . 6 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) ↔ (((𝑅 FrSe 𝐴𝑥𝐴) ∧ 𝜒′) ∧ 𝜂))
2 bnj908.4 . . . . . . . . . . 11 (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3 bnj908.10 . . . . . . . . . . 11 (𝜑′[𝑚 / 𝑛]𝜑)
4 bnj908.11 . . . . . . . . . . 11 (𝜓′[𝑚 / 𝑛]𝜓)
5 bnj908.12 . . . . . . . . . . 11 (𝜒′[𝑚 / 𝑛]𝜒)
6 vex 3352 . . . . . . . . . . 11 𝑚 ∈ V
72, 3, 4, 5, 6bnj207 31283 . . . . . . . . . 10 (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))
87biimpi 206 . . . . . . . . 9 (𝜒′ → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))
9 euex 2641 . . . . . . . . 9 (∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′) → ∃𝑓(𝑓 Fn 𝑚𝜑′𝜓′))
108, 9syl6 35 . . . . . . . 8 (𝜒′ → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))
1110impcom 394 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ 𝜒′) → ∃𝑓(𝑓 Fn 𝑚𝜑′𝜓′))
12 bnj908.17 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
1311, 12bnj1198 31198 . . . . . 6 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ 𝜒′) → ∃𝑓𝜏)
141, 13bnj832 31160 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓𝜏)
15 bnj645 31152 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → 𝜂)
16 19.41v 2028 . . . . 5 (∃𝑓(𝜏𝜂) ↔ (∃𝑓𝜏𝜂))
1714, 15, 16sylanbrc 564 . . . 4 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓(𝜏𝜂))
18 bnj642 31150 . . . 4 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → 𝑅 FrSe 𝐴)
19 19.41v 2028 . . . 4 (∃𝑓((𝜏𝜂) ∧ 𝑅 FrSe 𝐴) ↔ (∃𝑓(𝜏𝜂) ∧ 𝑅 FrSe 𝐴))
2017, 18, 19sylanbrc 564 . . 3 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓((𝜏𝜂) ∧ 𝑅 FrSe 𝐴))
21 bnj170 31098 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂) ↔ ((𝜏𝜂) ∧ 𝑅 FrSe 𝐴))
2220, 21bnj1198 31198 . 2 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓(𝑅 FrSe 𝐴𝜏𝜂))
23 bnj908.18 . . . 4 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
24 bnj908.19 . . . 4 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
25 bnj908.1 . . . . . 6 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2625, 3, 6bnj523 31289 . . . . 5 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
27 bnj908.2 . . . . . 6 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
2827, 4, 6bnj539 31293 . . . . 5 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
29 bnj908.3 . . . . 5 𝐷 = (ω ∖ {∅})
30 bnj908.16 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
3126, 28, 29, 30, 12, 23bnj544 31296 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
3223, 24, 31bnj561 31305 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
33 bnj908.13 . . . . . 6 (𝜑″[𝐺 / 𝑓]𝜑)
3430bnj528 31291 . . . . . 6 𝐺 ∈ V
3525, 33, 34bnj609 31319 . . . . 5 (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
3626, 29, 30, 12, 23, 31, 35bnj545 31297 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)
3723, 24, 36bnj562 31306 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜑″)
38 bnj908.20 . . . 4 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
39 bnj908.22 . . . 4 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
40 bnj908.23 . . . 4 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
41 bnj908.24 . . . 4 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
42 bnj908.25 . . . 4 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
43 bnj908.26 . . . 4 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
44 bnj908.21 . . . 4 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
45 bnj908.14 . . . . 5 (𝜓″[𝐺 / 𝑓]𝜓)
4627, 45, 34bnj611 31320 . . . 4 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
4729, 30, 12, 23, 24, 38, 39, 40, 41, 42, 43, 26, 28, 31, 44, 32, 46bnj571 31308 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜓″)
4832, 37, 473jca 1121 . 2 ((𝑅 FrSe 𝐴𝜏𝜂) → (𝐺 Fn 𝑛𝜑″𝜓″))
4922, 48bnj593 31147 1 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓(𝐺 Fn 𝑛𝜑″𝜓″))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wex 1851  wcel 2144  ∃!weu 2617  wne 2942  wral 3060  [wsbc 3585  cdif 3718  cun 3719  c0 4061  {csn 4314  cop 4320   ciun 4652   class class class wbr 4784   E cep 5161  suc csuc 5868   Fn wfn 6026  cfv 6031  ωcom 7211  w-bnj17 31086   predc-bnj14 31088   FrSe w-bnj15 31092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095  ax-reg 8652
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  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-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7212  df-bnj17 31087  df-bnj14 31089  df-bnj13 31091  df-bnj15 31093
This theorem is referenced by: (None)
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