Theorem fthi

Description: The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017)

	Ref	Expression
Hypotheses	isfth.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	isfth.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	isfth.j	⊢ 𝐽 = ( Hom ‘ 𝐷 )
	fthf1.f	⊢ ( 𝜑 → 𝐹 ( 𝐶 Faith 𝐷 ) 𝐺 )
	fthf1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	fthf1.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	fthi.r	⊢ ( 𝜑 → 𝑅 ∈ ( 𝑋 𝐻 𝑌 ) )
	fthi.s	⊢ ( 𝜑 → 𝑆 ∈ ( 𝑋 𝐻 𝑌 ) )
Assertion	fthi	⊢ ( 𝜑 → ( ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑅 ) = ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑆 ) ↔ 𝑅 = 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		isfth.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		isfth.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
3		isfth.j	⊢ 𝐽 = ( Hom ‘ 𝐷 )
4		fthf1.f	⊢ ( 𝜑 → 𝐹 ( 𝐶 Faith 𝐷 ) 𝐺 )
5		fthf1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		fthf1.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		fthi.r	⊢ ( 𝜑 → 𝑅 ∈ ( 𝑋 𝐻 𝑌 ) )
8		fthi.s	⊢ ( 𝜑 → 𝑆 ∈ ( 𝑋 𝐻 𝑌 ) )
9	1 2 3 4 5 6	fthf1	⊢ ( 𝜑 → ( 𝑋 𝐺 𝑌 ) : ( 𝑋 𝐻 𝑌 ) –1-1→ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) )
10		f1fveq	⊢ ( ( ( 𝑋 𝐺 𝑌 ) : ( 𝑋 𝐻 𝑌 ) –1-1→ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) ∧ ( 𝑅 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑆 ∈ ( 𝑋 𝐻 𝑌 ) ) ) → ( ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑅 ) = ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑆 ) ↔ 𝑅 = 𝑆 ) )
11	9 7 8 10	syl12anc	⊢ ( 𝜑 → ( ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑅 ) = ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑆 ) ↔ 𝑅 = 𝑆 ) )