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Theorem bnj908 32424
 Description: Technical lemma for bnj69 32503. 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 32191 . . . . . 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 3414 . . . . . . . . . . 11 𝑚 ∈ V
72, 3, 4, 5, 6bnj207 32374 . . . . . . . . . 10 (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))
87biimpi 219 . . . . . . . . 9 (𝜒′ → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))
9 euex 2597 . . . . . . . . 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 412 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ 𝜒′) → ∃𝑓(𝑓 Fn 𝑚𝜑′𝜓′))
12 bnj908.17 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
1311, 12bnj1198 32288 . . . . . 6 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ 𝜒′) → ∃𝑓𝜏)
141, 13bnj832 32250 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓𝜏)
15 bnj645 32242 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → 𝜂)
16 19.41v 1951 . . . . 5 (∃𝑓(𝜏𝜂) ↔ (∃𝑓𝜏𝜂))
1714, 15, 16sylanbrc 587 . . . 4 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓(𝜏𝜂))
18 bnj642 32240 . . . 4 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → 𝑅 FrSe 𝐴)
19 19.41v 1951 . . . 4 (∃𝑓((𝜏𝜂) ∧ 𝑅 FrSe 𝐴) ↔ (∃𝑓(𝜏𝜂) ∧ 𝑅 FrSe 𝐴))
2017, 18, 19sylanbrc 587 . . 3 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓((𝜏𝜂) ∧ 𝑅 FrSe 𝐴))
21 bnj170 32189 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂) ↔ ((𝜏𝜂) ∧ 𝑅 FrSe 𝐴))
2220, 21bnj1198 32288 . 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 32380 . . . . 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 32384 . . . . 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 32387 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
3223, 24, 31bnj561 32396 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
33 bnj908.13 . . . . . 6 (𝜑″[𝐺 / 𝑓]𝜑)
3430bnj528 32382 . . . . . 6 𝐺 ∈ V
3525, 33, 34bnj609 32410 . . . . 5 (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
3626, 29, 30, 12, 23, 31, 35bnj545 32388 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)
3723, 24, 36bnj562 32397 . . 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 32411 . . . 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 32399 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜓″)
4832, 37, 473jca 1126 . 2 ((𝑅 FrSe 𝐴𝜏𝜂) → (𝐺 Fn 𝑛𝜑″𝜓″))
4922, 48bnj593 32237 1 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓(𝐺 Fn 𝑛𝜑″𝜓″))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539  ∃wex 1782   ∈ wcel 2112  ∃!weu 2588   ≠ wne 2952  ∀wral 3071  [wsbc 3697   ∖ cdif 3856   ∪ cun 3857  ∅c0 4226  {csn 4523  ⟨cop 4529  ∪ ciun 4884   class class class wbr 5033   E cep 5435  suc csuc 6172   Fn wfn 6331  ‘cfv 6336  ωcom 7580   ∧ w-bnj17 32177   predc-bnj14 32179   FrSe w-bnj15 32183 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5299  ax-un 7460  ax-reg 9082 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-om 7581  df-bnj17 32178  df-bnj14 32180  df-bnj13 32182  df-bnj15 32184 This theorem is referenced by: (None)
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