Theorem comfval2 17737

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 2764	. 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 17726	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
11	8, 10	eleqtrd 2866	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
12		comfval2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
13	9, 2, 3, 6, 7	homfval 17726	. . 3 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍))
14	12, 13	eleqtrd 2866	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍))
15	1, 2, 3, 4, 5, 6, 7, 11, 14	comfval 17734	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144  ⟨cop 4590  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  Hom chom 17299  compcco 17300  Hom_f chomf 17700  comp_fccomf 17701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-homf 17704  df-comf 17705

This theorem is referenced by: (None)