Theorem comfval2 17628

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

Hypotheses

Ref	Expression
comfffval2.o	⊢ 𝑂 = (compf‘𝐶)
comfffval2.b	⊢ 𝐵 = (Base‘𝐶)
comfffval2.h	⊢ 𝐻 = (Homf ‘𝐶)
comfffval2.x	⊢ · = (comp‘𝐶)
comffval2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
comffval2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
comffval2.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
comfval2.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
comfval2.g	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))

Assertion

Ref	Expression
comfval2	⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))

Proof of Theorem comfval2

Step	Hyp	Ref	Expression
1		comfffval2.o	. 2 ⊢ 𝑂 = (compf‘𝐶)
2		comfffval2.b	. 2 ⊢ 𝐵 = (Base‘𝐶)
3		eqid 2736	. 2 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		comfffval2.x	. 2 ⊢ · = (comp‘𝐶)
5		comffval2.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		comffval2.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		comffval2.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
8		comfval2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
9		comfffval2.h	. . . 4 ⊢ 𝐻 = (Homf ‘𝐶)
10	9, 2, 3, 5, 6	homfval 17617	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
11	8, 10	eleqtrd 2838	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
12		comfval2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
13	9, 2, 3, 6, 7	homfval 17617	. . 3 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍))
14	12, 13	eleqtrd 2838	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍))
15	1, 2, 3, 4, 5, 6, 7, 11, 14	comfval 17625	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⟨cop 4586  ‘cfv 6492  (class class class)co 7358  Basecbs 17138  Hom chom 17190  compcco 17191  Hom_f chomf 17591  comp_fccomf 17592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-homf 17595  df-comf 17596

This theorem is referenced by: (None)