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Theorem cofu2 17880
Description: Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
cofuval.b 𝐵 = (Base‘𝐶)
cofuval.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
cofuval.g (𝜑𝐺 ∈ (𝐷 Func 𝐸))
cofu2nd.x (𝜑𝑋𝐵)
cofu2nd.y (𝜑𝑌𝐵)
cofu2.h 𝐻 = (Hom ‘𝐶)
cofu2.y (𝜑𝑅 ∈ (𝑋𝐻𝑌))
Assertion
Ref Expression
cofu2 (𝜑 → ((𝑋(2nd ‘(𝐺func 𝐹))𝑌)‘𝑅) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌))‘((𝑋(2nd𝐹)𝑌)‘𝑅)))

Proof of Theorem cofu2
StepHypRef Expression
1 cofuval.b . . . 4 𝐵 = (Base‘𝐶)
2 cofuval.f . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
3 cofuval.g . . . 4 (𝜑𝐺 ∈ (𝐷 Func 𝐸))
4 cofu2nd.x . . . 4 (𝜑𝑋𝐵)
5 cofu2nd.y . . . 4 (𝜑𝑌𝐵)
61, 2, 3, 4, 5cofu2nd 17879 . . 3 (𝜑 → (𝑋(2nd ‘(𝐺func 𝐹))𝑌) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)))
76fveq1d 6898 . 2 (𝜑 → ((𝑋(2nd ‘(𝐺func 𝐹))𝑌)‘𝑅) = (((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌))‘𝑅))
8 cofu2.h . . . 4 𝐻 = (Hom ‘𝐶)
9 eqid 2725 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
10 relfunc 17856 . . . . 5 Rel (𝐶 Func 𝐷)
11 1st2ndbr 8047 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1210, 2, 11sylancr 585 . . . 4 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
131, 8, 9, 12, 4, 5funcf2 17862 . . 3 (𝜑 → (𝑋(2nd𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1st𝐹)‘𝑋)(Hom ‘𝐷)((1st𝐹)‘𝑌)))
14 cofu2.y . . 3 (𝜑𝑅 ∈ (𝑋𝐻𝑌))
15 fvco3 6996 . . 3 (((𝑋(2nd𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1st𝐹)‘𝑋)(Hom ‘𝐷)((1st𝐹)‘𝑌)) ∧ 𝑅 ∈ (𝑋𝐻𝑌)) → (((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌))‘𝑅) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌))‘((𝑋(2nd𝐹)𝑌)‘𝑅)))
1613, 14, 15syl2anc 582 . 2 (𝜑 → (((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌))‘𝑅) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌))‘((𝑋(2nd𝐹)𝑌)‘𝑅)))
177, 16eqtrd 2765 1 (𝜑 → ((𝑋(2nd ‘(𝐺func 𝐹))𝑌)‘𝑅) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌))‘((𝑋(2nd𝐹)𝑌)‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098   class class class wbr 5149  ccom 5682  Rel wrel 5683  wf 6545  cfv 6549  (class class class)co 7419  1st c1st 7992  2nd c2nd 7993  Basecbs 17188  Hom chom 17252   Func cfunc 17848  func ccofu 17850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847  df-ixp 8917  df-func 17852  df-cofu 17854
This theorem is referenced by:  cofucl  17882  1st2ndprf  18205  uncf2  18237  yonedalem22  18278
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