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Theorem comfval2 17412
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)
comfval2.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
comfval2.g (𝜑𝐺 ∈ (𝑌𝐻𝑍))
Assertion
Ref Expression
comfval2 (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))

Proof of Theorem comfval2
StepHypRef Expression
1 comfffval2.o . 2 𝑂 = (compf𝐶)
2 comfffval2.b . 2 𝐵 = (Base‘𝐶)
3 eqid 2738 . 2 (Hom ‘𝐶) = (Hom ‘𝐶)
4 comfffval2.x . 2 · = (comp‘𝐶)
5 comffval2.x . 2 (𝜑𝑋𝐵)
6 comffval2.y . 2 (𝜑𝑌𝐵)
7 comffval2.z . 2 (𝜑𝑍𝐵)
8 comfval2.f . . 3 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
9 comfffval2.h . . . 4 𝐻 = (Homf𝐶)
109, 2, 3, 5, 6homfval 17401 . . 3 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
118, 10eleqtrd 2841 . 2 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
12 comfval2.g . . 3 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
139, 2, 3, 6, 7homfval 17401 . . 3 (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍))
1412, 13eleqtrd 2841 . 2 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍))
151, 2, 3, 4, 5, 6, 7, 11, 14comfval 17409 1 (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  cop 4567  cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973  compcco 16974  Homf chomf 17375  compfccomf 17376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-homf 17379  df-comf 17380
This theorem is referenced by: (None)
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