Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj601 Structured version   Visualization version   GIF version

Theorem bnj601 30733
Description: Technical lemma for bnj852 30734. 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
bnj601.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj601.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj601.3 𝐷 = (ω ∖ {∅})
bnj601.4 (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
bnj601.5 (𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))
Assertion
Ref Expression
bnj601 (𝑛 ≠ 1𝑜 → ((𝑛𝐷𝜃) → 𝜒))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐷,𝑓,𝑖   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑥,𝑓,𝑚,𝑛   𝜑,𝑖,𝑚   𝜓,𝑚
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜒(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜃(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝐴(𝑥)   𝐷(𝑥,𝑦,𝑚,𝑛)   𝑅(𝑥)

Proof of Theorem bnj601
Dummy variables 𝑝 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj601.1 . 2 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2 bnj601.2 . 2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj601.3 . 2 𝐷 = (ω ∖ {∅})
4 bnj601.4 . 2 (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
5 bnj601.5 . 2 (𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))
6 biid 251 . 2 ([𝑚 / 𝑛]𝜑[𝑚 / 𝑛]𝜑)
7 biid 251 . 2 ([𝑚 / 𝑛]𝜓[𝑚 / 𝑛]𝜓)
8 biid 251 . 2 ([𝑚 / 𝑛]𝜒[𝑚 / 𝑛]𝜒)
9 bnj602 30728 . . . . . . 7 (𝑦 = 𝑧 → pred(𝑦, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
109cbviunv 4530 . . . . . 6 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)
1110opeq2i 4379 . . . . 5 𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩ = ⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩
1211sneqi 4164 . . . 4 {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩} = {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}
1312uneq2i 3747 . . 3 (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩})
14 dfsbcq 3423 . . 3 ((𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) → ([(𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜑[(𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜑))
1513, 14ax-mp 5 . 2 ([(𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜑[(𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜑)
16 dfsbcq 3423 . . 3 ((𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) → ([(𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜓[(𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜓))
1713, 16ax-mp 5 . 2 ([(𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜓[(𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜓)
18 dfsbcq 3423 . . 3 ((𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) → ([(𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜒[(𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜒))
1913, 18ax-mp 5 . 2 ([(𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) / 𝑓]𝜒[(𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) / 𝑓]𝜒)
2013eqcomi 2630 . 2 (𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩}) = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
21 biid 251 . 2 ((𝑓 Fn 𝑚[𝑚 / 𝑛]𝜑[𝑚 / 𝑛]𝜓) ↔ (𝑓 Fn 𝑚[𝑚 / 𝑛]𝜑[𝑚 / 𝑛]𝜓))
22 biid 251 . 2 ((𝑚𝐷𝑛 = suc 𝑚𝑝𝑚) ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
23 biid 251 . 2 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
24 biid 251 . 2 ((𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))
25 biid 251 . 2 ((𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖) ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))
26 eqid 2621 . 2 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
27 eqid 2621 . 2 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)
28 eqid 2621 . 2 𝑦 ∈ ((𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩})‘𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ ((𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩})‘𝑖) pred(𝑦, 𝐴, 𝑅)
29 eqid 2621 . 2 𝑦 ∈ ((𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩})‘𝑝) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ ((𝑓 ∪ {⟨𝑚, 𝑧 ∈ (𝑓𝑝) pred(𝑧, 𝐴, 𝑅)⟩})‘𝑝) pred(𝑦, 𝐴, 𝑅)
301, 2, 3, 4, 5, 6, 7, 8, 15, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 20bnj600 30732 1 (𝑛 ≠ 1𝑜 → ((𝑛𝐷𝜃) → 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  ∃!weu 2469  wne 2790  wral 2907  [wsbc 3421  cdif 3556  cun 3557  c0 3896  {csn 4153  cop 4159   ciun 4490   class class class wbr 4618   E cep 4988  suc csuc 5689   Fn wfn 5847  cfv 5852  ωcom 7019  1𝑜c1o 7505  w-bnj17 30494   predc-bnj14 30496   FrSe w-bnj15 30500
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-reg 8449
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-1o 7512  df-bnj17 30495  df-bnj14 30497  df-bnj13 30499  df-bnj15 30501
This theorem is referenced by:  bnj852  30734
  Copyright terms: Public domain W3C validator