Theorem bnj96 34858

Description: Technical lemma for bnj150 34869. 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 𝐴 ∧ 𝑥 ∈ 𝐴) → dom 𝐹 = 1o)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem bnj96

Step	Hyp	Ref	Expression
1		bnj93 34856	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
2		dmsnopg 6235	. . 3 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → dom {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} = {∅})
3	1, 2	syl 17	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → dom {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} = {∅})
4		bnj96.1	. . 3 ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
5	4	dmeqi 5918	. 2 ⊢ dom 𝐹 = dom {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
6		df1o2 8512	. 2 ⊢ 1o = {∅}
7	3, 5, 6	3eqtr4g 2800	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → dom 𝐹 = 1o)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∅c0 4339  {csn 4631  ⟨cop 4637  dom cdm 5689  1_oc1o 8498   predc-bnj14 34681   FrSe w-bnj15 34685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-dm 5699  df-suc 6392  df-1o 8505  df-bnj13 34684  df-bnj15 34686

This theorem is referenced by:  bnj150  34869