Theorem fulli

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

	Ref	Expression
Hypotheses	isfull.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	isfull.j	⊢ 𝐽 = ( Hom ‘ 𝐷 )
	isfull.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	fullfo.f	⊢ ( 𝜑 → 𝐹 ( 𝐶 Full 𝐷 ) 𝐺 )
	fullfo.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	fullfo.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	fulli.r	⊢ ( 𝜑 → 𝑅 ∈ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) )
Assertion	fulli	⊢ ( 𝜑 → ∃ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) 𝑅 = ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑓 ) )

Proof

Step	Hyp	Ref	Expression
1		isfull.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		isfull.j	⊢ 𝐽 = ( Hom ‘ 𝐷 )
3		isfull.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
4		fullfo.f	⊢ ( 𝜑 → 𝐹 ( 𝐶 Full 𝐷 ) 𝐺 )
5		fullfo.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		fullfo.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		fulli.r	⊢ ( 𝜑 → 𝑅 ∈ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) )
8	1 2 3 4 5 6	fullfo	⊢ ( 𝜑 → ( 𝑋 𝐺 𝑌 ) : ( 𝑋 𝐻 𝑌 ) –onto→ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) )
9		foelrn	⊢ ( ( ( 𝑋 𝐺 𝑌 ) : ( 𝑋 𝐻 𝑌 ) –onto→ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) ∧ 𝑅 ∈ ( ( 𝐹 ‘ 𝑋 ) 𝐽 ( 𝐹 ‘ 𝑌 ) ) ) → ∃ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) 𝑅 = ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑓 ) )
10	8 7 9	syl2anc	⊢ ( 𝜑 → ∃ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) 𝑅 = ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑓 ) )