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Theorem comfval 16183
 Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval.o 𝑂 = (compf𝐶)
comfffval.b 𝐵 = (Base‘𝐶)
comfffval.h 𝐻 = (Hom ‘𝐶)
comfffval.x · = (comp‘𝐶)
comffval.x (𝜑𝑋𝐵)
comffval.y (𝜑𝑌𝐵)
comffval.z (𝜑𝑍𝐵)
comfval.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
comfval.g (𝜑𝐺 ∈ (𝑌𝐻𝑍))
Assertion
Ref Expression
comfval (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))

Proof of Theorem comfval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 comfffval.o . . 3 𝑂 = (compf𝐶)
2 comfffval.b . . 3 𝐵 = (Base‘𝐶)
3 comfffval.h . . 3 𝐻 = (Hom ‘𝐶)
4 comfffval.x . . 3 · = (comp‘𝐶)
5 comffval.x . . 3 (𝜑𝑋𝐵)
6 comffval.y . . 3 (𝜑𝑌𝐵)
7 comffval.z . . 3 (𝜑𝑍𝐵)
81, 2, 3, 4, 5, 6, 7comffval 16182 . 2 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
9 oveq12 6558 . . 3 ((𝑔 = 𝐺𝑓 = 𝐹) → (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
109adantl 481 . 2 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
11 comfval.g . 2 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
12 comfval.f . 2 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
13 ovex 6577 . . 3 (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ V
1413a1i 11 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ V)
158, 10, 11, 12, 14ovmpt2d 6686 1 (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  compcco 15780  compfccomf 16151 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-comf 16155 This theorem is referenced by:  comfval2  16186  comfeqval  16191
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