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Theorem bnj97 31254
Description: Technical lemma for bnj150 31264. 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 31251 . . 3 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
2 0ex 4984 . . . . 5 ∅ ∈ V
32bnj519 31123 . . . 4 ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
4 bnj96.1 . . . . 5 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
54funeqi 6118 . . . 4 (Fun 𝐹 ↔ Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
63, 5sylibr 225 . . 3 ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun 𝐹)
71, 6syl 17 . 2 ((𝑅 FrSe 𝐴𝑥𝐴) → Fun 𝐹)
8 opex 5122 . . . 4 ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ V
98snid 4402 . . 3 ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
109, 4eleqtrri 2884 . 2 ⟨∅, pred(𝑥, 𝐴, 𝑅)⟩ ∈ 𝐹
11 funopfv 6451 . 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 384   = wceq 1637  wcel 2156  Vcvv 3391  c0 4116  {csn 4370  cop 4376  Fun wfun 6091  cfv 6097   predc-bnj14 31075   FrSe w-bnj15 31079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6060  df-fun 6099  df-fv 6105  df-bnj13 31078  df-bnj15 31080
This theorem is referenced by:  bnj150  31264
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