Theorem comfval 17650

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 17649	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))
9		oveq12 7413	. . 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 7439	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ V)
14	8, 10, 11, 12, 13	ovmpod 7555	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  ⟨cop 4629  ‘cfv 6536  (class class class)co 7404  Basecbs 17150  Hom chom 17214  compcco 17215  comp_fccomf 17617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-comf 17621

This theorem is referenced by:  comfval2  17653  comfeqval  17658