Theorem comfval2 17717

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 2734	. 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 17706	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
11	8, 10	eleqtrd 2835	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
12		comfval2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
13	9, 2, 3, 6, 7	homfval 17706	. . 3 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍))
14	12, 13	eleqtrd 2835	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍))
15	1, 2, 3, 4, 5, 6, 7, 11, 14	comfval 17714	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ⟨cop 4612  ‘cfv 6541  (class class class)co 7413  Basecbs 17229  Hom chom 17284  compcco 17285  Hom_f chomf 17680  comp_fccomf 17681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997  df-homf 17684  df-comf 17685

This theorem is referenced by: (None)