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Theorem bnj600 32191
Description: Technical lemma for bnj852 32193. 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
bnj600.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj600.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj600.3 𝐷 = (ω ∖ {∅})
bnj600.4 (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
bnj600.5 (𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))
bnj600.10 (𝜑′[𝑚 / 𝑛]𝜑)
bnj600.11 (𝜓′[𝑚 / 𝑛]𝜓)
bnj600.12 (𝜒′[𝑚 / 𝑛]𝜒)
bnj600.13 (𝜑″[𝐺 / 𝑓]𝜑)
bnj600.14 (𝜓″[𝐺 / 𝑓]𝜓)
bnj600.15 (𝜒″[𝐺 / 𝑓]𝜒)
bnj600.16 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj600.17 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj600.18 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj600.19 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj600.20 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
bnj600.21 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
bnj600.22 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
bnj600.23 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
bnj600.24 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
bnj600.25 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
bnj600.26 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
Assertion
Ref Expression
bnj600 (𝑛 ≠ 1o → ((𝑛𝐷𝜃) → 𝜒))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑝   𝑦,𝐴,𝑓,𝑖,𝑛,𝑝   𝐷,𝑓,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,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 bnj600
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bnj600.5 . . . . . 6 (𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))
2 bnj600.13 . . . . . 6 (𝜑″[𝐺 / 𝑓]𝜑)
3 bnj600.14 . . . . . 6 (𝜓″[𝐺 / 𝑓]𝜓)
4 bnj600.17 . . . . . 6 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
5 bnj600.19 . . . . . 6 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
6 bnj600.16 . . . . . . 7 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
76bnj528 32161 . . . . . 6 𝐺 ∈ V
8 bnj600.4 . . . . . . 7 (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
9 bnj600.10 . . . . . . 7 (𝜑′[𝑚 / 𝑛]𝜑)
10 bnj600.11 . . . . . . 7 (𝜓′[𝑚 / 𝑛]𝜓)
11 bnj600.12 . . . . . . 7 (𝜒′[𝑚 / 𝑛]𝜒)
12 vex 3497 . . . . . . 7 𝑚 ∈ V
138, 9, 10, 11, 12bnj207 32153 . . . . . 6 (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))
14 bnj600.1 . . . . . . 7 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
1514, 2, 7bnj609 32189 . . . . . 6 (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
16 bnj600.2 . . . . . . 7 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
1716, 3, 7bnj611 32190 . . . . . 6 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
18 bnj600.3 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
1918bnj168 32000 . . . . . . . . 9 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚𝐷 𝑛 = suc 𝑚)
20 df-rex 3144 . . . . . . . . 9 (∃𝑚𝐷 𝑛 = suc 𝑚 ↔ ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
2119, 20sylib 220 . . . . . . . 8 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
2218bnj158 31999 . . . . . . . . . . . . . 14 (𝑚𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
23 df-rex 3144 . . . . . . . . . . . . . 14 (∃𝑝 ∈ ω 𝑚 = suc 𝑝 ↔ ∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
2422, 23sylib 220 . . . . . . . . . . . . 13 (𝑚𝐷 → ∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
2524adantr 483 . . . . . . . . . . . 12 ((𝑚𝐷𝑛 = suc 𝑚) → ∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
2625ancri 552 . . . . . . . . . . 11 ((𝑚𝐷𝑛 = suc 𝑚) → (∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚𝐷𝑛 = suc 𝑚)))
2726bnj534 32010 . . . . . . . . . 10 ((𝑚𝐷𝑛 = suc 𝑚) → ∃𝑝((𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚𝐷𝑛 = suc 𝑚)))
28 bnj432 31986 . . . . . . . . . . 11 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ((𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚𝐷𝑛 = suc 𝑚)))
2928exbii 1844 . . . . . . . . . 10 (∃𝑝(𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ∃𝑝((𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚𝐷𝑛 = suc 𝑚)))
3027, 29sylibr 236 . . . . . . . . 9 ((𝑚𝐷𝑛 = suc 𝑚) → ∃𝑝(𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
3130eximi 1831 . . . . . . . 8 (∃𝑚(𝑚𝐷𝑛 = suc 𝑚) → ∃𝑚𝑝(𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
3221, 31syl 17 . . . . . . 7 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚𝑝(𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
3352exbii 1845 . . . . . . 7 (∃𝑚𝑝𝜂 ↔ ∃𝑚𝑝(𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
3432, 33sylibr 236 . . . . . 6 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚𝑝𝜂)
35 rsp 3205 . . . . . . . . 9 (∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒) → (𝑚𝐷 → (𝑚 E 𝑛[𝑚 / 𝑛]𝜒)))
361, 35sylbi 219 . . . . . . . 8 (𝜃 → (𝑚𝐷 → (𝑚 E 𝑛[𝑚 / 𝑛]𝜒)))
37363imp 1107 . . . . . . 7 ((𝜃𝑚𝐷𝑚 E 𝑛) → [𝑚 / 𝑛]𝜒)
3837, 11sylibr 236 . . . . . 6 ((𝜃𝑚𝐷𝑚 E 𝑛) → 𝜒′)
39 bnj600.18 . . . . . . 7 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
4014, 9, 12bnj523 32159 . . . . . . . 8 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
4116, 10, 12bnj539 32163 . . . . . . . 8 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
4240, 41, 18, 6, 4, 39bnj544 32166 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
4339, 5, 42bnj561 32175 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
4440, 18, 6, 4, 39, 42, 15bnj545 32167 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)
4539, 5, 44bnj562 32176 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜑″)
46 bnj600.20 . . . . . . 7 (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
47 bnj600.22 . . . . . . 7 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
48 bnj600.23 . . . . . . 7 𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
49 bnj600.24 . . . . . . 7 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
50 bnj600.25 . . . . . . 7 𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)
51 bnj600.26 . . . . . . 7 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
52 bnj600.21 . . . . . . 7 (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
5318, 6, 4, 39, 5, 46, 47, 48, 49, 50, 51, 40, 41, 42, 52, 43, 17bnj571 32178 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜓″)
54 biid 263 . . . . . 6 ([𝑧 / 𝑓]𝜑[𝑧 / 𝑓]𝜑)
55 biid 263 . . . . . 6 ([𝑧 / 𝑓]𝜓[𝑧 / 𝑓]𝜓)
56 biid 263 . . . . . 6 ([𝐺 / 𝑧][𝑧 / 𝑓]𝜑[𝐺 / 𝑧][𝑧 / 𝑓]𝜑)
57 biid 263 . . . . . 6 ([𝐺 / 𝑧][𝑧 / 𝑓]𝜓[𝐺 / 𝑧][𝑧 / 𝑓]𝜓)
581, 2, 3, 4, 5, 7, 13, 15, 17, 34, 38, 43, 45, 53, 14, 16, 54, 55, 56, 57bnj607 32188 . . . . 5 ((𝑛 ≠ 1o𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 𝑛𝜑𝜓)))
5914, 16, 18bnj579 32186 . . . . . . 7 (𝑛𝐷 → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))
6059a1d 25 . . . . . 6 (𝑛𝐷 → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓)))
61603ad2ant2 1130 . . . . 5 ((𝑛 ≠ 1o𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓)))
6258, 61jcad 515 . . . 4 ((𝑛 ≠ 1o𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → (∃𝑓(𝑓 Fn 𝑛𝜑𝜓) ∧ ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))))
63 df-eu 2650 . . . 4 (∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ (∃𝑓(𝑓 Fn 𝑛𝜑𝜓) ∧ ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓)))
6462, 63syl6ibr 254 . . 3 ((𝑛 ≠ 1o𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
6564, 8sylibr 236 . 2 ((𝑛 ≠ 1o𝑛𝐷𝜃) → 𝜒)
66653expib 1118 1 (𝑛 ≠ 1o → ((𝑛𝐷𝜃) → 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wex 1776  wcel 2110  ∃*wmo 2616  ∃!weu 2649  wne 3016  wral 3138  wrex 3139  [wsbc 3771  cdif 3932  cun 3933  c0 4290  {csn 4566  cop 4572   ciun 4918   class class class wbr 5065   E cep 5463  suc csuc 6192   Fn wfn 6349  cfv 6354  ωcom 7579  1oc1o 8094  w-bnj17 31956   predc-bnj14 31958   FrSe w-bnj15 31962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-reg 9055
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-om 7580  df-1o 8101  df-bnj17 31957  df-bnj14 31959  df-bnj13 31961  df-bnj15 31963
This theorem is referenced by:  bnj601  32192
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