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Theorem cofu2 16312
 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 16311 . . 3 (𝜑 → (𝑋(2nd ‘(𝐺func 𝐹))𝑌) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)))
76fveq1d 6087 . 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 2606 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
10 relfunc 16288 . . . . 5 Rel (𝐶 Func 𝐷)
11 1st2ndbr 7082 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1210, 2, 11sylancr 693 . . . 4 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
131, 8, 9, 12, 4, 5funcf2 16294 . . 3 (𝜑 → (𝑋(2nd𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1st𝐹)‘𝑋)(Hom ‘𝐷)((1st𝐹)‘𝑌)))
14 cofu2.y . . 3 (𝜑𝑅 ∈ (𝑋𝐻𝑌))
15 fvco3 6167 . . 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 690 . 2 (𝜑 → (((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌))‘𝑅) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌))‘((𝑋(2nd𝐹)𝑌)‘𝑅)))
177, 16eqtrd 2640 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 1474   ∈ wcel 1976   class class class wbr 4574   ∘ ccom 5029  Rel wrel 5030  ⟶wf 5783  ‘cfv 5787  (class class class)co 6524  1st c1st 7031  2nd c2nd 7032  Basecbs 15638  Hom chom 15722   Func cfunc 16280   ∘func ccofu 16282 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-1st 7033  df-2nd 7034  df-map 7720  df-ixp 7769  df-func 16284  df-cofu 16286 This theorem is referenced by:  cofucl  16314  1st2ndprf  16612  uncf2  16643  yonedalem22  16684
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