Metamath Proof Explorer


Theorem bnj96

Description: Technical lemma for bnj150 . 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.)

Ref Expression
Hypothesis bnj96.1 𝐹 = { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ }
Assertion bnj96 ( ( 𝑅 FrSe 𝐴𝑥𝐴 ) → dom 𝐹 = 1o )

Proof

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