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Theorem cofu2 17147
 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 17146 . . 3 (𝜑 → (𝑋(2nd ‘(𝐺func 𝐹))𝑌) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)))
76fveq1d 6654 . 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 2822 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
10 relfunc 17123 . . . . 5 Rel (𝐶 Func 𝐷)
11 1st2ndbr 7727 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1210, 2, 11sylancr 590 . . . 4 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
131, 8, 9, 12, 4, 5funcf2 17129 . . 3 (𝜑 → (𝑋(2nd𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1st𝐹)‘𝑋)(Hom ‘𝐷)((1st𝐹)‘𝑌)))
14 cofu2.y . . 3 (𝜑𝑅 ∈ (𝑋𝐻𝑌))
15 fvco3 6742 . . 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 587 . 2 (𝜑 → (((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌))‘𝑅) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌))‘((𝑋(2nd𝐹)𝑌)‘𝑅)))
177, 16eqtrd 2857 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 1538   ∈ wcel 2114   class class class wbr 5042   ∘ ccom 5536  Rel wrel 5537  ⟶wf 6330  ‘cfv 6334  (class class class)co 7140  1st c1st 7673  2nd c2nd 7674  Basecbs 16474  Hom chom 16567   Func cfunc 17115   ∘func ccofu 17117 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-map 8395  df-ixp 8449  df-func 17119  df-cofu 17121 This theorem is referenced by:  cofucl  17149  1st2ndprf  17447  uncf2  17478  yonedalem22  17519
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