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Theorem bnj150 32376
 Description: Technical lemma for bnj151 32377. 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 bnj150.8 . . . 4 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
21bnj95 32364 . . 3 𝐹 ∈ V
3 sbceq1a 3707 . . . 4 (𝑓 = 𝐹 → (𝜁′[𝐹 / 𝑓]𝜁′))
4 bnj150.11 . . . 4 (𝜁″[𝐹 / 𝑓]𝜁′)
53, 4bitr4di 292 . . 3 (𝑓 = 𝐹 → (𝜁′𝜁″))
6 0ex 5177 . . . . . . . . 9 ∅ ∈ V
7 bnj93 32363 . . . . . . . . 9 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
8 funsng 6386 . . . . . . . . 9 ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
96, 7, 8sylancr 590 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
101funeqi 6356 . . . . . . . 8 (Fun 𝐹 ↔ Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
119, 10sylibr 237 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴) → Fun 𝐹)
121bnj96 32365 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴) → dom 𝐹 = 1o)
1311, 12bnj1422 32337 . . . . . 6 ((𝑅 FrSe 𝐴𝑥𝐴) → 𝐹 Fn 1o)
141bnj97 32366 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
15 bnj150.1 . . . . . . . 8 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
16 bnj150.4 . . . . . . . 8 (𝜑′[1o / 𝑛]𝜑)
17 bnj150.9 . . . . . . . 8 (𝜑″[𝐹 / 𝑓]𝜑′)
1815, 16, 17, 1bnj125 32372 . . . . . . 7 (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
1914, 18sylibr 237 . . . . . 6 ((𝑅 FrSe 𝐴𝑥𝐴) → 𝜑″)
2013, 19jca 515 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1o𝜑″))
21 bnj98 32367 . . . . . 6 𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
22 bnj150.2 . . . . . . 7 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
23 bnj150.5 . . . . . . 7 (𝜓′[1o / 𝑛]𝜓)
24 bnj150.10 . . . . . . 7 (𝜓″[𝐹 / 𝑓]𝜓′)
2522, 23, 24, 1bnj126 32373 . . . . . 6 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
2621, 25mpbir 234 . . . . 5 𝜓″
27 df-3an 1086 . . . . 5 ((𝐹 Fn 1o𝜑″𝜓″) ↔ ((𝐹 Fn 1o𝜑″) ∧ 𝜓″))
2820, 26, 27sylanblrc 593 . . . 4 ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1o𝜑″𝜓″))
29 bnj150.3 . . . . . 6 (𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
30 bnj150.7 . . . . . 6 (𝜁′[1o / 𝑛]𝜁)
3129, 30, 16, 23bnj121 32370 . . . . 5 (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)))
321, 17, 24, 4, 31bnj124 32371 . . . 4 (𝜁″ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹 Fn 1o𝜑″𝜓″)))
3328, 32mpbir 234 . . 3 𝜁″
342, 5, 33ceqsexv2d 3459 . 2 𝑓𝜁′
35 bnj150.6 . . . 4 (𝜃0 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1o𝜑′𝜓′)))
36 19.37v 1998 . . . 4 (∃𝑓((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃𝑓(𝑓 Fn 1o𝜑′𝜓′)))
3735, 36bitr4i 281 . . 3 (𝜃0 ↔ ∃𝑓((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)))
3837, 31bnj133 32225 . 2 (𝜃0 ↔ ∃𝑓𝜁′)
3934, 38mpbir 234 1 𝜃0
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3070  Vcvv 3409  [wsbc 3696  ∅c0 4225  {csn 4522  ⟨cop 4528  ∪ ciun 4883  suc csuc 6171  Fun wfun 6329   Fn wfn 6330  ‘cfv 6335  ωcom 7579  1oc1o 8105   predc-bnj14 32186   FrSe w-bnj15 32190 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-fv 6343  df-1o 8112  df-bnj13 32189  df-bnj15 32191 This theorem is referenced by:  bnj151  32377
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