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Theorem bnj96 32558
Description: Technical lemma for bnj150 32569. 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 32556 . . 3 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
2 dmsnopg 6076 . . 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 5773 . 2 dom 𝐹 = dom {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
6 df1o2 8214 . 2 1o = {∅}
73, 5, 63eqtr4g 2803 1 ((𝑅 FrSe 𝐴𝑥𝐴) → dom 𝐹 = 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  Vcvv 3408  c0 4237  {csn 4541  cop 4547  dom cdm 5551  1oc1o 8195   predc-bnj14 32379   FrSe w-bnj15 32383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-dm 5561  df-suc 6219  df-1o 8202  df-bnj13 32382  df-bnj15 32384
This theorem is referenced by:  bnj150  32569
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