Theorem bnj96 35027

Description: Technical lemma for bnj150 35038. 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 35025	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
2		dmsnopg 6173	. . 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 5855	. 2 ⊢ dom 𝐹 = dom {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
6		df1o2 8407	. 2 ⊢ 1o = {∅}
7	3, 5, 6	3eqtr4g 2797	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → dom 𝐹 = 1o)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ∅c0 4274  {csn 4568  ⟨cop 4574  dom cdm 5626  1_oc1o 8393   predc-bnj14 34851   FrSe w-bnj15 34855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-dm 5636  df-suc 6325  df-1o 8400  df-bnj13 34854  df-bnj15 34856

This theorem is referenced by:  bnj150  35038