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Theorem comfval 17674
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 17673 . 2 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
9 oveq12 7424 . . 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 ovexd 7450 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ V)
148, 10, 11, 12, 13ovmpod 7568 1 (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  Vcvv 3470  cop 4631  cfv 6543  (class class class)co 7415  Basecbs 17174  Hom chom 17238  compcco 17239  compfccomf 17641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7988  df-2nd 7989  df-comf 17645
This theorem is referenced by:  comfval2  17677  comfeqval  17682
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