Theorem coeq2

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

	Ref	Expression
Assertion	coeq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∘ 𝐴 ) = ( 𝐶 ∘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		coss2	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∘ 𝐴 ) ⊆ ( 𝐶 ∘ 𝐵 ) )
2		coss2	⊢ ( 𝐵 ⊆ 𝐴 → ( 𝐶 ∘ 𝐵 ) ⊆ ( 𝐶 ∘ 𝐴 ) )
3	1 2	anim12i	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) → ( ( 𝐶 ∘ 𝐴 ) ⊆ ( 𝐶 ∘ 𝐵 ) ∧ ( 𝐶 ∘ 𝐵 ) ⊆ ( 𝐶 ∘ 𝐴 ) ) )
4		eqss	⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) )
5		eqss	⊢ ( ( 𝐶 ∘ 𝐴 ) = ( 𝐶 ∘ 𝐵 ) ↔ ( ( 𝐶 ∘ 𝐴 ) ⊆ ( 𝐶 ∘ 𝐵 ) ∧ ( 𝐶 ∘ 𝐵 ) ⊆ ( 𝐶 ∘ 𝐴 ) ) )
6	3 4 5	3imtr4i	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∘ 𝐴 ) = ( 𝐶 ∘ 𝐵 ) )