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Theorem bnj908 30255
 Description: Technical lemma for bnj69 30332. 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 30019 . . . . . 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 3176 . . . . . . . . . . 11 𝑚 ∈ V
72, 3, 4, 5, 6bnj207 30205 . . . . . . . . . 10 (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))
87biimpi 205 . . . . . . . . 9 (𝜒′ → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))
9 euex 2482 . . . . . . . . 9 (∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′) → ∃𝑓(𝑓 Fn 𝑚𝜑′𝜓′))
108, 9syl6 34 . . . . . . . 8 (𝜒′ → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))
1110impcom 445 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ 𝜒′) → ∃𝑓(𝑓 Fn 𝑚𝜑′𝜓′))
12 bnj908.17 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
1311, 12bnj1198 30120 . . . . . 6 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ 𝜒′) → ∃𝑓𝜏)
141, 13bnj832 30082 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓𝜏)
15 bnj645 30074 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → 𝜂)
16 19.41v 1901 . . . . 5 (∃𝑓(𝜏𝜂) ↔ (∃𝑓𝜏𝜂))
1714, 15, 16sylanbrc 695 . . . 4 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓(𝜏𝜂))
18 bnj642 30072 . . . 4 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → 𝑅 FrSe 𝐴)
19 19.41v 1901 . . . 4 (∃𝑓((𝜏𝜂) ∧ 𝑅 FrSe 𝐴) ↔ (∃𝑓(𝜏𝜂) ∧ 𝑅 FrSe 𝐴))
2017, 18, 19sylanbrc 695 . . 3 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓((𝜏𝜂) ∧ 𝑅 FrSe 𝐴))
21 bnj170 30017 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂) ↔ ((𝜏𝜂) ∧ 𝑅 FrSe 𝐴))
2220, 21bnj1198 30120 . 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 30211 . . . . 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 30215 . . . . 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 30218 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
3223, 24, 31bnj561 30227 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
33 bnj908.13 . . . . . 6 (𝜑″[𝐺 / 𝑓]𝜑)
3430bnj528 30213 . . . . . 6 𝐺 ∈ V
3525, 33, 34bnj609 30241 . . . . 5 (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
3626, 29, 30, 12, 23, 31, 35bnj545 30219 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)
3723, 24, 36bnj562 30228 . . 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 30242 . . . 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 30230 . . 3 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜓″)
4832, 37, 473jca 1235 . 2 ((𝑅 FrSe 𝐴𝜏𝜂) → (𝐺 Fn 𝑛𝜑″𝜓″))
4922, 48bnj593 30069 1 ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓(𝐺 Fn 𝑛𝜑″𝜓″))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458   ≠ wne 2780  ∀wral 2896  [wsbc 3402   ∖ cdif 3537   ∪ cun 3538  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ ciun 4455   class class class wbr 4583   E cep 4947  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957   ∧ w-bnj17 30005   predc-bnj14 30007   FrSe w-bnj15 30011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012 This theorem is referenced by: (None)
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