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Theorem bnj97 33865
Description: Technical lemma for bnj150 33875. 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.)
Hypothesis
Ref Expression
bnj96.1 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
Assertion
Ref Expression
bnj97 ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem bnj97
StepHypRef Expression
1 bnj93 33862 . . 3 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
2 0ex 5306 . . . . 5 ∅ ∈ V
32bnj519 33735 . . . 4 ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
4 bnj96.1 . . . . 5 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
54funeqi 6566 . . . 4 (Fun 𝐹 ↔ Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
63, 5sylibr 233 . . 3 ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun 𝐹)
71, 6syl 17 . 2 ((𝑅 FrSe 𝐴𝑥𝐴) → Fun 𝐹)
8 opex 5463 . . . 4 ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ V
98snid 4663 . . 3 ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
109, 4eleqtrri 2832 . 2 ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ 𝐹
11 funopfv 6940 . 2 (Fun 𝐹 → (⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ 𝐹 → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)))
127, 10, 11mpisyl 21 1 ((𝑅 FrSe 𝐴𝑥𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  c0 4321  {csn 4627  cop 4633  Fun wfun 6534  cfv 6540   predc-bnj14 33687   FrSe w-bnj15 33691
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-bnj13 33690  df-bnj15 33692
This theorem is referenced by:  bnj150  33875
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