Theorem cofu2 17601

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	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
cofuval.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
cofu2nd.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
cofu2nd.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
cofu2.h	⊢ 𝐻 = (Hom ‘𝐶)
cofu2.y	⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
cofu2	⊢ (𝜑 → ((𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌)‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)))

Proof of Theorem cofu2

Step	Hyp	Ref	Expression
1		cofuval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		cofuval.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		cofuval.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
4		cofu2nd.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		cofu2nd.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	1, 2, 3, 4, 5	cofu2nd 17600	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)))
7	6	fveq1d 6776	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌)‘𝑅) = (((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌))‘𝑅))
8		cofu2.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
9		eqid 2738	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10		relfunc 17577	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
11		1st2ndbr 7883	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
12	10, 2, 11	sylancr 587	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
13	1, 8, 9, 12, 4, 5	funcf2 17583	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1st ‘𝐹)‘𝑋)(Hom ‘𝐷)((1st ‘𝐹)‘𝑌)))
14		cofu2.y	. . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))
15		fvco3 6867	. . 3 ⊢ (((𝑋(2nd ‘𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1st ‘𝐹)‘𝑋)(Hom ‘𝐷)((1st ‘𝐹)‘𝑌)) ∧ 𝑅 ∈ (𝑋𝐻𝑌)) → (((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌))‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)))
16	13, 14, 15	syl2anc 584	. 2 ⊢ (𝜑 → (((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌))‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)))
17	7, 16	eqtrd 2778	1 ⊢ (𝜑 → ((𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌)‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106   class class class wbr 5074   ∘ ccom 5593  Rel wrel 5594  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  Basecbs 16912  Hom chom 16973   Func cfunc 17569   ∘_func ccofu 17571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-ixp 8686  df-func 17573  df-cofu 17575

This theorem is referenced by:  cofucl  17603  1st2ndprf  17923  uncf2  17955  yonedalem22  17996