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Theorem bnj97 31750
Description: Technical lemma for bnj150 31760. 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 31747 . . 3 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
2 0ex 5109 . . . . 5 ∅ ∈ V
32bnj519 31619 . . . 4 ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
4 bnj96.1 . . . . 5 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
54funeqi 6253 . . . 4 (Fun 𝐹 ↔ Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
63, 5sylibr 235 . . 3 ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun 𝐹)
71, 6syl 17 . 2 ((𝑅 FrSe 𝐴𝑥𝐴) → Fun 𝐹)
8 opex 5255 . . . 4 ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ V
98snid 4512 . . 3 ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
109, 4eleqtrri 2884 . 2 ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ 𝐹
11 funopfv 6592 . 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 1525  wcel 2083  Vcvv 3440  c0 4217  {csn 4478  cop 4484  Fun wfun 6226  cfv 6232   predc-bnj14 31571   FrSe w-bnj15 31575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-iota 6196  df-fun 6234  df-fv 6240  df-bnj13 31574  df-bnj15 31576
This theorem is referenced by:  bnj150  31760
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