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Theorem cofu2 17871
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 17870 . . 3 (𝜑 → (𝑋(2nd ‘(𝐺func 𝐹))𝑌) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)))
76fveq1d 6899 . 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 2728 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
10 relfunc 17847 . . . . 5 Rel (𝐶 Func 𝐷)
11 1st2ndbr 8046 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1210, 2, 11sylancr 586 . . . 4 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
131, 8, 9, 12, 4, 5funcf2 17853 . . 3 (𝜑 → (𝑋(2nd𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1st𝐹)‘𝑋)(Hom ‘𝐷)((1st𝐹)‘𝑌)))
14 cofu2.y . . 3 (𝜑𝑅 ∈ (𝑋𝐻𝑌))
15 fvco3 6997 . . 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 583 . 2 (𝜑 → (((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌))‘𝑅) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌))‘((𝑋(2nd𝐹)𝑌)‘𝑅)))
177, 16eqtrd 2768 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 1534  wcel 2099   class class class wbr 5148  ccom 5682  Rel wrel 5683  wf 6544  cfv 6548  (class class class)co 7420  1st c1st 7991  2nd c2nd 7992  Basecbs 17179  Hom chom 17243   Func cfunc 17839  func ccofu 17841
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
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 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  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 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-map 8846  df-ixp 8916  df-func 17843  df-cofu 17845
This theorem is referenced by:  cofucl  17873  1st2ndprf  18196  uncf2  18228  yonedalem22  18269
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