Theorem bnj96 30640

Description: Technical lemma for bnj150 30651. 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 30638	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
2		dmsnopg 5565	. . 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 5285	. 2 ⊢ dom 𝐹 = dom {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
6		df1o2 7517	. 2 ⊢ 1𝑜 = {∅}
7	3, 5, 6	3eqtr4g 2680	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → dom 𝐹 = 1𝑜)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186  ∅c0 3891  {csn 4148  ⟨cop 4154  dom cdm 5074  1_𝑜c1o 7498   predc-bnj14 30458   FrSe w-bnj15 30462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-dm 5084  df-suc 5688  df-1o 7505  df-bnj13 30461  df-bnj15 30463

This theorem is referenced by:  bnj150  30651