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Metamath Proof Explorer

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 ) )