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Theorem bnj96 31242
Description: Technical lemma for bnj150 31253. 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.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj96.1 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
Assertion
Ref Expression
bnj96 ((𝑅 FrSe 𝐴𝑥𝐴) → dom 𝐹 = 1𝑜)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem bnj96
StepHypRef Expression
1 bnj93 31240 . . 3 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
2 dmsnopg 5765 . . 3 ( pred(𝑥, 𝐴, 𝑅) ∈ V → dom {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} = {∅})
31, 2syl 17 . 2 ((𝑅 FrSe 𝐴𝑥𝐴) → dom {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} = {∅})
4 bnj96.1 . . 3 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
54dmeqi 5480 . 2 dom 𝐹 = dom {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
6 df1o2 7741 . 2 1𝑜 = {∅}
73, 5, 63eqtr4g 2819 1 ((𝑅 FrSe 𝐴𝑥𝐴) → dom 𝐹 = 1𝑜)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  c0 4058  {csn 4321  cop 4327  dom cdm 5266  1𝑜c1o 7722   predc-bnj14 31063   FrSe w-bnj15 31067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-dm 5276  df-suc 5890  df-1o 7729  df-bnj13 31066  df-bnj15 31068
This theorem is referenced by:  bnj150  31253
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