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Theorem bnj600 31217
Description: Technical lemma for bnj852 31219. 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 31187 . . . . . 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 3307 . . . . . . 7 𝑚 ∈ V
138, 9, 10, 11, 12bnj207 31179 . . . . . 6 (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))
14 bnj600.1 . . . . . . 7 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
1514, 2, 7bnj609 31215 . . . . . 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 31216 . . . . . 6 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
18 bnj600.3 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
1918bnj168 31026 . . . . . . . . 9 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚𝐷 𝑛 = suc 𝑚)
20 df-rex 3020 . . . . . . . . 9 (∃𝑚𝐷 𝑛 = suc 𝑚 ↔ ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
2119, 20sylib 208 . . . . . . . 8 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
2218bnj158 31025 . . . . . . . . . . . . . 14 (𝑚𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
23 df-rex 3020 . . . . . . . . . . . . . 14 (∃𝑝 ∈ ω 𝑚 = suc 𝑝 ↔ ∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
2422, 23sylib 208 . . . . . . . . . . . . 13 (𝑚𝐷 → ∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
2524adantr 472 . . . . . . . . . . . 12 ((𝑚𝐷𝑛 = suc 𝑚) → ∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
2625ancri 576 . . . . . . . . . . 11 ((𝑚𝐷𝑛 = suc 𝑚) → (∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚𝐷𝑛 = suc 𝑚)))
2726bnj534 31036 . . . . . . . . . 10 ((𝑚𝐷𝑛 = suc 𝑚) → ∃𝑝((𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚𝐷𝑛 = suc 𝑚)))
28 bnj432 31012 . . . . . . . . . . 11 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ((𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚𝐷𝑛 = suc 𝑚)))
2928exbii 1887 . . . . . . . . . 10 (∃𝑝(𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ∃𝑝((𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚𝐷𝑛 = suc 𝑚)))
3027, 29sylibr 224 . . . . . . . . 9 ((𝑚𝐷𝑛 = suc 𝑚) → ∃𝑝(𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
3130eximi 1875 . . . . . . . 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 1888 . . . . . . 7 (∃𝑚𝑝𝜂 ↔ ∃𝑚𝑝(𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
3432, 33sylibr 224 . . . . . 6 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚𝑝𝜂)
35 rsp 3031 . . . . . . . . 9 (∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒) → (𝑚𝐷 → (𝑚 E 𝑛[𝑚 / 𝑛]𝜒)))
361, 35sylbi 207 . . . . . . . 8 (𝜃 → (𝑚𝐷 → (𝑚 E 𝑛[𝑚 / 𝑛]𝜒)))
37363imp 1101 . . . . . . 7 ((𝜃𝑚𝐷𝑚 E 𝑛) → [𝑚 / 𝑛]𝜒)
3837, 11sylibr 224 . . . . . 6 ((𝜃𝑚𝐷𝑚 E 𝑛) → 𝜒′)
39 bnj600.18 . . . . . . 7 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
4014, 9, 12bnj523 31185 . . . . . . . 8 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
4116, 10, 12bnj539 31189 . . . . . . . 8 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
4240, 41, 18, 6, 4, 39bnj544 31192 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
4339, 5, 42bnj561 31201 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
4440, 18, 6, 4, 39, 42, 15bnj545 31193 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)
4539, 5, 44bnj562 31202 . . . . . 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 31204 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝜓″)
54 biid 251 . . . . . 6 ([𝑧 / 𝑓]𝜑[𝑧 / 𝑓]𝜑)
55 biid 251 . . . . . 6 ([𝑧 / 𝑓]𝜓[𝑧 / 𝑓]𝜓)
56 biid 251 . . . . . 6 ([𝐺 / 𝑧][𝑧 / 𝑓]𝜑[𝐺 / 𝑧][𝑧 / 𝑓]𝜑)
57 biid 251 . . . . . 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 31214 . . . . 5 ((𝑛 ≠ 1𝑜𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 𝑛𝜑𝜓)))
5914, 16, 18bnj579 31212 . . . . . . 7 (𝑛𝐷 → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))
6059a1d 25 . . . . . 6 (𝑛𝐷 → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓)))
61603ad2ant2 1126 . . . . 5 ((𝑛 ≠ 1𝑜𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓)))
6258, 61jcad 556 . . . 4 ((𝑛 ≠ 1𝑜𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → (∃𝑓(𝑓 Fn 𝑛𝜑𝜓) ∧ ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))))
63 eu5 2597 . . . 4 (∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ (∃𝑓(𝑓 Fn 𝑛𝜑𝜓) ∧ ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓)))
6462, 63syl6ibr 242 . . 3 ((𝑛 ≠ 1𝑜𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
6564, 8sylibr 224 . 2 ((𝑛 ≠ 1𝑜𝑛𝐷𝜃) → 𝜒)
66653expib 1116 1 (𝑛 ≠ 1𝑜 → ((𝑛𝐷𝜃) → 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wex 1817  wcel 2103  ∃!weu 2571  ∃*wmo 2572  wne 2896  wral 3014  wrex 3015  [wsbc 3541  cdif 3677  cun 3678  c0 4023  {csn 4285  cop 4291   ciun 4628   class class class wbr 4760   E cep 5132  suc csuc 5838   Fn wfn 5996  cfv 6001  ωcom 7182  1𝑜c1o 7673  w-bnj17 30982   predc-bnj14 30984   FrSe w-bnj15 30988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-reg 8613
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-om 7183  df-1o 7680  df-bnj17 30983  df-bnj14 30985  df-bnj13 30987  df-bnj15 30989
This theorem is referenced by:  bnj601  31218
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