Theorem isexid

Description: The predicate G has a left and right identity element. (Contributed by FL, 2-Nov-2009) (Revised by Mario Carneiro, 22-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Hypothesis	isexid.1	⊢ 𝑋 = dom dom 𝐺
Assertion	isexid	⊢ ( 𝐺 ∈ 𝐴 → ( 𝐺 ∈ ExId ↔ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		isexid.1	⊢ 𝑋 = dom dom 𝐺
2		dmeq	⊢ ( 𝑔 = 𝐺 → dom 𝑔 = dom 𝐺 )
3	2	dmeqd	⊢ ( 𝑔 = 𝐺 → dom dom 𝑔 = dom dom 𝐺 )
4	3 1	eqtr4di	⊢ ( 𝑔 = 𝐺 → dom dom 𝑔 = 𝑋 )
5		oveq	⊢ ( 𝑔 = 𝐺 → ( 𝑥 𝑔 𝑦 ) = ( 𝑥 𝐺 𝑦 ) )
6	5	eqeq1d	⊢ ( 𝑔 = 𝐺 → ( ( 𝑥 𝑔 𝑦 ) = 𝑦 ↔ ( 𝑥 𝐺 𝑦 ) = 𝑦 ) )
7		oveq	⊢ ( 𝑔 = 𝐺 → ( 𝑦 𝑔 𝑥 ) = ( 𝑦 𝐺 𝑥 ) )
8	7	eqeq1d	⊢ ( 𝑔 = 𝐺 → ( ( 𝑦 𝑔 𝑥 ) = 𝑦 ↔ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) )
9	6 8	anbi12d	⊢ ( 𝑔 = 𝐺 → ( ( ( 𝑥 𝑔 𝑦 ) = 𝑦 ∧ ( 𝑦 𝑔 𝑥 ) = 𝑦 ) ↔ ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) )
10	4 9	raleqbidv	⊢ ( 𝑔 = 𝐺 → ( ∀ 𝑦 ∈ dom dom 𝑔 ( ( 𝑥 𝑔 𝑦 ) = 𝑦 ∧ ( 𝑦 𝑔 𝑥 ) = 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) )
11	4 10	rexeqbidv	⊢ ( 𝑔 = 𝐺 → ( ∃ 𝑥 ∈ dom dom 𝑔 ∀ 𝑦 ∈ dom dom 𝑔 ( ( 𝑥 𝑔 𝑦 ) = 𝑦 ∧ ( 𝑦 𝑔 𝑥 ) = 𝑦 ) ↔ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) )
12		df-exid	⊢ ExId = { 𝑔 ∣ ∃ 𝑥 ∈ dom dom 𝑔 ∀ 𝑦 ∈ dom dom 𝑔 ( ( 𝑥 𝑔 𝑦 ) = 𝑦 ∧ ( 𝑦 𝑔 𝑥 ) = 𝑦 ) }
13	11 12	elab2g	⊢ ( 𝐺 ∈ 𝐴 → ( 𝐺 ∈ ExId ↔ ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐺 𝑥 ) = 𝑦 ) ) )