Metamath Proof Explorer


Theorem comfval2

Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017)

Ref Expression
Hypotheses comfffval2.o O = comp 𝑓 C
comfffval2.b B = Base C
comfffval2.h H = Hom 𝑓 C
comfffval2.x · ˙ = comp C
comffval2.x φ X B
comffval2.y φ Y B
comffval2.z φ Z B
comfval2.f φ F X H Y
comfval2.g φ G Y H Z
Assertion comfval2 φ G X Y O Z F = G X Y · ˙ Z F

Proof

Step Hyp Ref Expression
1 comfffval2.o O = comp 𝑓 C
2 comfffval2.b B = Base C
3 comfffval2.h H = Hom 𝑓 C
4 comfffval2.x · ˙ = comp C
5 comffval2.x φ X B
6 comffval2.y φ Y B
7 comffval2.z φ Z B
8 comfval2.f φ F X H Y
9 comfval2.g φ G Y H Z
10 eqid Hom C = Hom C
11 3 2 10 5 6 homfval φ X H Y = X Hom C Y
12 8 11 eleqtrd φ F X Hom C Y
13 3 2 10 6 7 homfval φ Y H Z = Y Hom C Z
14 9 13 eleqtrd φ G Y Hom C Z
15 1 2 10 4 5 6 7 12 14 comfval φ G X Y O Z F = G X Y · ˙ Z F