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Theorem comffval2 16185
 Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval2.o 𝑂 = (compf𝐶)
comfffval2.b 𝐵 = (Base‘𝐶)
comfffval2.h 𝐻 = (Homf𝐶)
comfffval2.x · = (comp‘𝐶)
comffval2.x (𝜑𝑋𝐵)
comffval2.y (𝜑𝑌𝐵)
comffval2.z (𝜑𝑍𝐵)
Assertion
Ref Expression
comffval2 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
Distinct variable groups:   𝑓,𝑔,𝐵   𝐶,𝑓,𝑔   · ,𝑓,𝑔   𝑓,𝑋,𝑔   𝑓,𝑌,𝑔   𝜑,𝑓,𝑔   𝑓,𝑍,𝑔
Allowed substitution hints:   𝐻(𝑓,𝑔)   𝑂(𝑓,𝑔)

Proof of Theorem comffval2
StepHypRef Expression
1 comfffval2.o . . 3 𝑂 = (compf𝐶)
2 comfffval2.b . . 3 𝐵 = (Base‘𝐶)
3 eqid 2610 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
4 comfffval2.x . . 3 · = (comp‘𝐶)
5 comffval2.x . . 3 (𝜑𝑋𝐵)
6 comffval2.y . . 3 (𝜑𝑌𝐵)
7 comffval2.z . . 3 (𝜑𝑍𝐵)
81, 2, 3, 4, 5, 6, 7comffval 16182 . 2 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑍), 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
9 comfffval2.h . . . 4 𝐻 = (Homf𝐶)
109, 2, 3, 6, 7homfval 16175 . . 3 (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍))
119, 2, 3, 5, 6homfval 16175 . . 3 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
12 eqidd 2611 . . 3 (𝜑 → (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓) = (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓))
1310, 11, 12mpt2eq123dv 6615 . 2 (𝜑 → (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)) = (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑍), 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
148, 13eqtr4d 2647 1 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  Hom chom 15779  compcco 15780  Homf chomf 16150  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-homf 16154  df-comf 16155 This theorem is referenced by: (None)
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