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Theorem comfval2 17077
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 17066 . . 3 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
118, 10eleqtrd 2835 . 2 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
12 comfval2.g . . 3 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
139, 2, 3, 6, 7homfval 17066 . . 3 (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍))
1412, 13eleqtrd 2835 . 2 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍))
151, 2, 3, 4, 5, 6, 7, 11, 14comfval 17074 1 (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  cop 4522  cfv 6339  (class class class)co 7170  Basecbs 16586  Hom chom 16679  compcco 16680  Homf chomf 17040  compfccomf 17041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-homf 17044  df-comf 17045
This theorem is referenced by: (None)
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