Theorem coeq1

Description: Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997)

	Ref	Expression
Assertion	coeq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∘ 𝐶 ) = ( 𝐵 ∘ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		coss1	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∘ 𝐶 ) ⊆ ( 𝐵 ∘ 𝐶 ) )
2		coss1	⊢ ( 𝐵 ⊆ 𝐴 → ( 𝐵 ∘ 𝐶 ) ⊆ ( 𝐴 ∘ 𝐶 ) )
3	1 2	anim12i	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) → ( ( 𝐴 ∘ 𝐶 ) ⊆ ( 𝐵 ∘ 𝐶 ) ∧ ( 𝐵 ∘ 𝐶 ) ⊆ ( 𝐴 ∘ 𝐶 ) ) )
4		eqss	⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) )
5		eqss	⊢ ( ( 𝐴 ∘ 𝐶 ) = ( 𝐵 ∘ 𝐶 ) ↔ ( ( 𝐴 ∘ 𝐶 ) ⊆ ( 𝐵 ∘ 𝐶 ) ∧ ( 𝐵 ∘ 𝐶 ) ⊆ ( 𝐴 ∘ 𝐶 ) ) )
6	3 4 5	3imtr4i	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∘ 𝐶 ) = ( 𝐵 ∘ 𝐶 ) )