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Theorem cofu2 16942
 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 16941 . . 3 (𝜑 → (𝑋(2nd ‘(𝐺func 𝐹))𝑌) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)))
76fveq1d 6450 . 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 2778 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
10 relfunc 16918 . . . . 5 Rel (𝐶 Func 𝐷)
11 1st2ndbr 7498 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1210, 2, 11sylancr 581 . . . 4 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
131, 8, 9, 12, 4, 5funcf2 16924 . . 3 (𝜑 → (𝑋(2nd𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1st𝐹)‘𝑋)(Hom ‘𝐷)((1st𝐹)‘𝑌)))
14 cofu2.y . . 3 (𝜑𝑅 ∈ (𝑋𝐻𝑌))
15 fvco3 6537 . . 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 579 . 2 (𝜑 → (((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌))‘𝑅) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌))‘((𝑋(2nd𝐹)𝑌)‘𝑅)))
177, 16eqtrd 2814 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 1601   ∈ wcel 2107   class class class wbr 4888   ∘ ccom 5361  Rel wrel 5362  ⟶wf 6133  ‘cfv 6137  (class class class)co 6924  1st c1st 7445  2nd c2nd 7446  Basecbs 16266  Hom chom 16360   Func cfunc 16910   ∘func ccofu 16912 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-map 8144  df-ixp 8197  df-func 16914  df-cofu 16916 This theorem is referenced by:  cofucl  16944  1st2ndprf  17243  uncf2  17274  yonedalem22  17315
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