Theorem coss2cnvepres

Description: Special case of coss1cnvres . (Contributed by Peter Mazsa, 8-Jun-2020)

	Ref	Expression
Assertion	coss2cnvepres	⊢ ≀ ◡ ( ◡ E ↾ 𝐴 ) = { ⟨ 𝑢 , 𝑣 ⟩ ∣ ( ( 𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ) ∧ ∃ 𝑥 ( 𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣 ) ) }

Proof

Step	Hyp	Ref	Expression
1		coss1cnvres	⊢ ≀ ◡ ( ◡ E ↾ 𝐴 ) = { ⟨ 𝑢 , 𝑣 ⟩ ∣ ( ( 𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ) ∧ ∃ 𝑥 ( 𝑢 ◡ E 𝑥 ∧ 𝑣 ◡ E 𝑥 ) ) }
2		brcnvep	⊢ ( 𝑢 ∈ V → ( 𝑢 ◡ E 𝑥 ↔ 𝑥 ∈ 𝑢 ) )
3	2	elv	⊢ ( 𝑢 ◡ E 𝑥 ↔ 𝑥 ∈ 𝑢 )
4		brcnvep	⊢ ( 𝑣 ∈ V → ( 𝑣 ◡ E 𝑥 ↔ 𝑥 ∈ 𝑣 ) )
5	4	elv	⊢ ( 𝑣 ◡ E 𝑥 ↔ 𝑥 ∈ 𝑣 )
6	3 5	anbi12i	⊢ ( ( 𝑢 ◡ E 𝑥 ∧ 𝑣 ◡ E 𝑥 ) ↔ ( 𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣 ) )
7	6	exbii	⊢ ( ∃ 𝑥 ( 𝑢 ◡ E 𝑥 ∧ 𝑣 ◡ E 𝑥 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣 ) )
8	7	anbi2i	⊢ ( ( ( 𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ) ∧ ∃ 𝑥 ( 𝑢 ◡ E 𝑥 ∧ 𝑣 ◡ E 𝑥 ) ) ↔ ( ( 𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ) ∧ ∃ 𝑥 ( 𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣 ) ) )
9	8	opabbii	⊢ { ⟨ 𝑢 , 𝑣 ⟩ ∣ ( ( 𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ) ∧ ∃ 𝑥 ( 𝑢 ◡ E 𝑥 ∧ 𝑣 ◡ E 𝑥 ) ) } = { ⟨ 𝑢 , 𝑣 ⟩ ∣ ( ( 𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ) ∧ ∃ 𝑥 ( 𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣 ) ) }
10	1 9	eqtri	⊢ ≀ ◡ ( ◡ E ↾ 𝐴 ) = { ⟨ 𝑢 , 𝑣 ⟩ ∣ ( ( 𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ) ∧ ∃ 𝑥 ( 𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣 ) ) }