Theorem gidval

Description: The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Hypothesis	gidval.1	⊢ 𝑋 = ran 𝐺
Assertion	gidval	⊢ ( 𝐺 ∈ 𝑉 → ( GId ‘ 𝐺 ) = ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		gidval.1	⊢ 𝑋 = ran 𝐺
2		elex	⊢ ( 𝐺 ∈ 𝑉 → 𝐺 ∈ V )
3		rneq	⊢ ( 𝑔 = 𝐺 → ran 𝑔 = ran 𝐺 )
4	3 1	eqtr4di	⊢ ( 𝑔 = 𝐺 → ran 𝑔 = 𝑋 )
5		oveq	⊢ ( 𝑔 = 𝐺 → ( 𝑢 𝑔 𝑥 ) = ( 𝑢 𝐺 𝑥 ) )
6	5	eqeq1d	⊢ ( 𝑔 = 𝐺 → ( ( 𝑢 𝑔 𝑥 ) = 𝑥 ↔ ( 𝑢 𝐺 𝑥 ) = 𝑥 ) )
7		oveq	⊢ ( 𝑔 = 𝐺 → ( 𝑥 𝑔 𝑢 ) = ( 𝑥 𝐺 𝑢 ) )
8	7	eqeq1d	⊢ ( 𝑔 = 𝐺 → ( ( 𝑥 𝑔 𝑢 ) = 𝑥 ↔ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) )
9	6 8	anbi12d	⊢ ( 𝑔 = 𝐺 → ( ( ( 𝑢 𝑔 𝑥 ) = 𝑥 ∧ ( 𝑥 𝑔 𝑢 ) = 𝑥 ) ↔ ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) )
10	4 9	raleqbidv	⊢ ( 𝑔 = 𝐺 → ( ∀ 𝑥 ∈ ran 𝑔 ( ( 𝑢 𝑔 𝑥 ) = 𝑥 ∧ ( 𝑥 𝑔 𝑢 ) = 𝑥 ) ↔ ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) )
11	4 10	riotaeqbidv	⊢ ( 𝑔 = 𝐺 → ( ℩ 𝑢 ∈ ran 𝑔 ∀ 𝑥 ∈ ran 𝑔 ( ( 𝑢 𝑔 𝑥 ) = 𝑥 ∧ ( 𝑥 𝑔 𝑢 ) = 𝑥 ) ) = ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) )
12		df-gid	⊢ GId = ( 𝑔 ∈ V ↦ ( ℩ 𝑢 ∈ ran 𝑔 ∀ 𝑥 ∈ ran 𝑔 ( ( 𝑢 𝑔 𝑥 ) = 𝑥 ∧ ( 𝑥 𝑔 𝑢 ) = 𝑥 ) ) )
13		riotaex	⊢ ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) ∈ V
14	11 12 13	fvmpt	⊢ ( 𝐺 ∈ V → ( GId ‘ 𝐺 ) = ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) )
15	2 14	syl	⊢ ( 𝐺 ∈ 𝑉 → ( GId ‘ 𝐺 ) = ( ℩ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ( 𝑥 𝐺 𝑢 ) = 𝑥 ) ) )