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 31072
Description: Technical lemma for bnj151 31073. 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 4823 . . . . . . . . . 10 ∅ ∈ V
2 bnj93 31059 . . . . . . . . . 10 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
3 funsng 5975 . . . . . . . . . 10 ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
41, 2, 3sylancr 696 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑥𝐴) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
5 bnj150.8 . . . . . . . . . 10 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
65funeqi 5947 . . . . . . . . 9 (Fun 𝐹 ↔ Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
74, 6sylibr 224 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → Fun 𝐹)
85bnj96 31061 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → dom 𝐹 = 1𝑜)
97, 8bnj1422 31034 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴) → 𝐹 Fn 1𝑜)
105bnj97 31062 . . . . . . . 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 31068 . . . . . . . 8 (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
1510, 14sylibr 224 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴) → 𝜑″)
169, 15jca 553 . . . . . 6 ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″))
17 bnj98 31063 . . . . . . 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 31069 . . . . . . 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 1056 . . . . 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 31066 . . . . 5 (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)))
305, 13, 20, 26, 29bnj124 31067 . . . 4 (𝜁″ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1𝑜𝜑″𝜓″)))
3125, 30mpbir 221 . . 3 𝜁″
325bnj95 31060 . . . 4 𝐹 ∈ V
33 sbceq1a 3479 . . . . 5 (𝑓 = 𝐹 → (𝜁′[𝐹 / 𝑓]𝜁′))
3433, 26syl6bbr 278 . . . 4 (𝑓 = 𝐹 → (𝜁′𝜁″))
3532, 34spcev 3331 . . 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 1966 . . . 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 30921 . 2 (𝜃0 ↔ ∃𝑓𝜁′)
4136, 40mpbir 221 1 𝜃0
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wral 2941  Vcvv 3231  [wsbc 3468  c0 3948  {csn 4210  cop 4216   ciun 4552  suc csuc 5763  Fun wfun 5920   Fn wfn 5921  cfv 5926  ωcom 7107  1𝑜c1o 7598   predc-bnj14 30882   FrSe w-bnj15 30886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-1o 7605  df-bnj13 30885  df-bnj15 30887
This theorem is referenced by:  bnj151  31073
  Copyright terms: Public domain W3C validator