Theorem comfval 17794

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 17793	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))
9		oveq12 7426	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
10	9	adantl 487	. 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 7452	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ V)
14	8, 10, 11, 12, 13	ovmpod 7569	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ⟨cop 4593  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  Hom chom 17359  compcco 17360  comp_fccomf 17761

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-comf 17765

This theorem is used by:  comfval2  17797  comfeqval  17802