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Theorem bnj600 30718
Description: Technical lemma for bnj852 30720. 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 30688 . . . . . 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 3189 . . . . . . 7 𝑚 ∈ V
138, 9, 10, 11, 12bnj207 30680 . . . . . 6 (𝜒′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑚𝜑′𝜓′)))
14 bnj600.1 . . . . . . 7 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
1514, 2, 7bnj609 30716 . . . . . 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 30717 . . . . . 6 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
18 bnj600.3 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
1918bnj168 30527 . . . . . . . . 9 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚𝐷 𝑛 = suc 𝑚)
20 df-rex 2913 . . . . . . . . 9 (∃𝑚𝐷 𝑛 = suc 𝑚 ↔ ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
2119, 20sylib 208 . . . . . . . 8 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
2218bnj158 30526 . . . . . . . . . . . . . 14 (𝑚𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
23 df-rex 2913 . . . . . . . . . . . . . 14 (∃𝑝 ∈ ω 𝑚 = suc 𝑝 ↔ ∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
2422, 23sylib 208 . . . . . . . . . . . . 13 (𝑚𝐷 → ∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
2524adantr 481 . . . . . . . . . . . 12 ((𝑚𝐷𝑛 = suc 𝑚) → ∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
2625ancri 574 . . . . . . . . . . 11 ((𝑚𝐷𝑛 = suc 𝑚) → (∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚𝐷𝑛 = suc 𝑚)))
2726bnj534 30537 . . . . . . . . . 10 ((𝑚𝐷𝑛 = suc 𝑚) → ∃𝑝((𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚𝐷𝑛 = suc 𝑚)))
28 bnj432 30510 . . . . . . . . . . 11 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ((𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚𝐷𝑛 = suc 𝑚)))
2928exbii 1771 . . . . . . . . . 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 1759 . . . . . . . 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 1772 . . . . . . 7 (∃𝑚𝑝𝜂 ↔ ∃𝑚𝑝(𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
3432, 33sylibr 224 . . . . . 6 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚𝑝𝜂)
35 rsp 2924 . . . . . . . . 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 1254 . . . . . . 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 30686 . . . . . . . 8 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
4116, 10, 12bnj539 30690 . . . . . . . 8 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
4240, 41, 18, 6, 4, 39bnj544 30693 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
4339, 5, 42bnj561 30702 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
4440, 18, 6, 4, 39, 42, 15bnj545 30694 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)
4539, 5, 44bnj562 30703 . . . . . 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 30705 . . . . . 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 30715 . . . . 5 ((𝑛 ≠ 1𝑜𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 𝑛𝜑𝜓)))
5914, 16, 18bnj579 30713 . . . . . . 7 (𝑛𝐷 → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))
6059a1d 25 . . . . . 6 (𝑛𝐷 → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓)))
61603ad2ant2 1081 . . . . 5 ((𝑛 ≠ 1𝑜𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓)))
6258, 61jcad 555 . . . 4 ((𝑛 ≠ 1𝑜𝑛𝐷𝜃) → ((𝑅 FrSe 𝐴𝑥𝐴) → (∃𝑓(𝑓 Fn 𝑛𝜑𝜓) ∧ ∃*𝑓(𝑓 Fn 𝑛𝜑𝜓))))
63 eu5 2495 . . . 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 1265 1 (𝑛 ≠ 1𝑜 → ((𝑛𝐷𝜃) → 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  ∃!weu 2469  ∃*wmo 2470  wne 2790  wral 2907  wrex 2908  [wsbc 3418  cdif 3553  cun 3554  c0 3893  {csn 4150  cop 4156   ciun 4487   class class class wbr 4615   E cep 4985  suc csuc 5686   Fn wfn 5844  cfv 5849  ωcom 7015  1𝑜c1o 7501  w-bnj17 30480   predc-bnj14 30482   FrSe w-bnj15 30486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-reg 8444
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-om 7016  df-1o 7508  df-bnj17 30481  df-bnj14 30483  df-bnj13 30485  df-bnj15 30487
This theorem is referenced by:  bnj601  30719
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