Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj150 Structured version   Visualization version   GIF version

Theorem bnj150 30707
Description: Technical lemma for bnj151 30708. 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
bnj150.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj150.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj150.3 (𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
bnj150.4 (𝜑′[1𝑜 / 𝑛]𝜑)
bnj150.5 (𝜓′[1𝑜 / 𝑛]𝜓)
bnj150.6 (𝜃0 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
bnj150.7 (𝜁′[1𝑜 / 𝑛]𝜁)
bnj150.8 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
bnj150.9 (𝜑″[𝐹 / 𝑓]𝜑′)
bnj150.10 (𝜓″[𝐹 / 𝑓]𝜓′)
bnj150.11 (𝜁″[𝐹 / 𝑓]𝜁′)
Assertion
Ref Expression
bnj150 𝜃0
Distinct variable groups:   𝐴,𝑓,𝑛,𝑥   𝑓,𝐹,𝑖,𝑦   𝑅,𝑓,𝑛,𝑥   𝑓,𝜁″   𝑖,𝑛,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜁(𝑥,𝑦,𝑓,𝑖,𝑛)   𝐴(𝑦,𝑖)   𝑅(𝑦,𝑖)   𝐹(𝑥,𝑛)   𝜑′(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜁′(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜑″(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓″(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜁″(𝑥,𝑦,𝑖,𝑛)   𝜃0(𝑥,𝑦,𝑓,𝑖,𝑛)

Proof of Theorem bnj150
StepHypRef Expression
1 0ex 4760 . . . . . . . . . 10 ∅ ∈ V
2 bnj93 30694 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
3 funsng 5905 . . . . . . . . . 10 ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
41, 2, 3sylancr 694 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑥𝐴) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
5 bnj150.8 . . . . . . . . . 10 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
65funeqi 5878 . . . . . . . . 9 (Fun 𝐹 ↔ Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
74, 6sylibr 224 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → Fun 𝐹)
85bnj96 30696 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → dom 𝐹 = 1𝑜)
97, 8bnj1422 30669 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴) → 𝐹 Fn 1𝑜)
105bnj97 30697 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
11 bnj150.1 . . . . . . . . 9 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
12 bnj150.4 . . . . . . . . 9 (𝜑′[1𝑜 / 𝑛]𝜑)
13 bnj150.9 . . . . . . . . 9 (𝜑″[𝐹 / 𝑓]𝜑′)
1411, 12, 13, 5bnj125 30703 . . . . . . . 8 (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
1510, 14sylibr 224 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴) → 𝜑″)
169, 15jca 554 . . . . . 6 ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″))
17 bnj98 30698 . . . . . . 7 𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
18 bnj150.2 . . . . . . . 8 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
19 bnj150.5 . . . . . . . 8 (𝜓′[1𝑜 / 𝑛]𝜓)
20 bnj150.10 . . . . . . . 8 (𝜓″[𝐹 / 𝑓]𝜓′)
2118, 19, 20, 5bnj126 30704 . . . . . . 7 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
2217, 21mpbir 221 . . . . . 6 𝜓″
2316, 22jctir 560 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴) → ((𝐹 Fn 1𝑜𝜑″) ∧ 𝜓″))
24 df-3an 1038 . . . . 5 ((𝐹 Fn 1𝑜𝜑″𝜓″) ↔ ((𝐹 Fn 1𝑜𝜑″) ∧ 𝜓″))
2523, 24sylibr 224 . . . 4 ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″𝜓″))
26 bnj150.11 . . . . 5 (𝜁″[𝐹 / 𝑓]𝜁′)
27 bnj150.3 . . . . . 6 (𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
28 bnj150.7 . . . . . 6 (𝜁′[1𝑜 / 𝑛]𝜁)
2927, 28, 12, 19bnj121 30701 . . . . 5 (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)))
305, 13, 20, 26, 29bnj124 30702 . . . 4 (𝜁″ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″𝜓″)))
3125, 30mpbir 221 . . 3 𝜁″
325bnj95 30695 . . . 4 𝐹 ∈ V
33 sbceq1a 3433 . . . . 5 (𝑓 = 𝐹 → (𝜁′[𝐹 / 𝑓]𝜁′))
3433, 26syl6bbr 278 . . . 4 (𝑓 = 𝐹 → (𝜁′𝜁″))
3532, 34spcev 3290 . . 3 (𝜁″ → ∃𝑓𝜁′)
3631, 35ax-mp 5 . 2 𝑓𝜁′
37 bnj150.6 . . . 4 (𝜃0 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
38 19.37v 1907 . . . 4 (∃𝑓((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
3937, 38bitr4i 267 . . 3 (𝜃0 ↔ ∃𝑓((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)))
4039, 29bnj133 30554 . 2 (𝜃0 ↔ ∃𝑓𝜁′)
4136, 40mpbir 221 1 𝜃0
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wral 2908  Vcvv 3190  [wsbc 3422  c0 3897  {csn 4155  cop 4161   ciun 4492  suc csuc 5694  Fun wfun 5851   Fn wfn 5852  cfv 5857  ωcom 7027  1𝑜c1o 7513   predc-bnj14 30514   FrSe w-bnj15 30518
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  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-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865  df-1o 7520  df-bnj13 30517  df-bnj15 30519
This theorem is referenced by:  bnj151  30708
  Copyright terms: Public domain W3C validator