Theorem comfval 17714

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

Hypotheses

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

Assertion

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

Proof of Theorem comfval

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		comfffval.o	. . 3 ⊢ 𝑂 = (compf‘𝐶)
2		comfffval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		comfffval.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
4		comfffval.x	. . 3 ⊢ · = (comp‘𝐶)
5		comffval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		comffval.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		comffval.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
8	1, 2, 3, 4, 5, 6, 7	comffval 17713	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))
9		oveq12 7422	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
10	9	adantl 481	. 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
11		comfval.g	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
12		comfval.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
13		ovexd 7448	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ V)
14	8, 10, 11, 12, 13	ovmpod 7567	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3463  ⟨cop 4612  ‘cfv 6541  (class class class)co 7413  Basecbs 17229  Hom chom 17284  compcco 17285  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-comf 17685

This theorem is referenced by:  comfval2  17717  comfeqval  17722