Theorem ffthf1o

Description: The morphism map of a fully faithful functor is a bijection. (Contributed by Mario Carneiro, 29-Jan-2017)

	Ref	Expression
Hypotheses	isfth.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	isfth.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	isfth.j	⊢ 𝐽 = ( Hom ‘ 𝐷 )
	ffthf1o.f	⊢ ( 𝜑 → 𝐹 ( ( 𝐶 Full 𝐷 ) ∩ ( 𝐶 Faith 𝐷 ) ) 𝐺 )
	ffthf1o.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ffthf1o.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	ffthf1o	⊢ ( 𝜑 → ( 𝑋 𝐺 𝑌 ) : ( 𝑋 𝐻 𝑌 ) –1-1-onto→ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		isfth.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		isfth.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
3		isfth.j	⊢ 𝐽 = ( Hom ‘ 𝐷 )
4		ffthf1o.f	⊢ ( 𝜑 → 𝐹 ( ( 𝐶 Full 𝐷 ) ∩ ( 𝐶 Faith 𝐷 ) ) 𝐺 )
5		ffthf1o.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		ffthf1o.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		brin	⊢ ( 𝐹 ( ( 𝐶 Full 𝐷 ) ∩ ( 𝐶 Faith 𝐷 ) ) 𝐺 ↔ ( 𝐹 ( 𝐶 Full 𝐷 ) 𝐺 ∧ 𝐹 ( 𝐶 Faith 𝐷 ) 𝐺 ) )
8	4 7	sylib	⊢ ( 𝜑 → ( 𝐹 ( 𝐶 Full 𝐷 ) 𝐺 ∧ 𝐹 ( 𝐶 Faith 𝐷 ) 𝐺 ) )
9	8	simprd	⊢ ( 𝜑 → 𝐹 ( 𝐶 Faith 𝐷 ) 𝐺 )
10	1 2 3 9 5 6	fthf1	⊢ ( 𝜑 → ( 𝑋 𝐺 𝑌 ) : ( 𝑋 𝐻 𝑌 ) –1-1→ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) )
11	8	simpld	⊢ ( 𝜑 → 𝐹 ( 𝐶 Full 𝐷 ) 𝐺 )
12	1 3 2 11 5 6	fullfo	⊢ ( 𝜑 → ( 𝑋 𝐺 𝑌 ) : ( 𝑋 𝐻 𝑌 ) –onto→ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) )
13		df-f1o	⊢ ( ( 𝑋 𝐺 𝑌 ) : ( 𝑋 𝐻 𝑌 ) –1-1-onto→ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) ↔ ( ( 𝑋 𝐺 𝑌 ) : ( 𝑋 𝐻 𝑌 ) –1-1→ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) ∧ ( 𝑋 𝐺 𝑌 ) : ( 𝑋 𝐻 𝑌 ) –onto→ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) ) )
14	10 12 13	sylanbrc	⊢ ( 𝜑 → ( 𝑋 𝐺 𝑌 ) : ( 𝑋 𝐻 𝑌 ) –1-1-onto→ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) )