Theorem dmeq

Description: Equality theorem for domain. (Contributed by NM, 11-Aug-1994)

	Ref	Expression
Assertion	dmeq	⊢ ( 𝐴 = 𝐵 → dom 𝐴 = dom 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		dmss	⊢ ( 𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵 )
2		dmss	⊢ ( 𝐵 ⊆ 𝐴 → dom 𝐵 ⊆ dom 𝐴 )
3	1 2	anim12i	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) → ( dom 𝐴 ⊆ dom 𝐵 ∧ dom 𝐵 ⊆ dom 𝐴 ) )
4		eqss	⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) )
5		eqss	⊢ ( dom 𝐴 = dom 𝐵 ↔ ( dom 𝐴 ⊆ dom 𝐵 ∧ dom 𝐵 ⊆ dom 𝐴 ) )
6	3 4 5	3imtr4i	⊢ ( 𝐴 = 𝐵 → dom 𝐴 = dom 𝐵 )