Theorem cofuval 17971

Description: Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
cofuval.b	⊢ 𝐵 = (Base‘𝐶)
cofuval.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
cofuval.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))

Assertion

Ref	Expression
cofuval	⊢ (𝜑 → (𝐺 ∘func 𝐹) = ⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩)

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥, 𝑦)   𝐷(𝑥, 𝑦)   𝐸(𝑥, 𝑦)

Proof of Theorem cofuval

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cofu 17949	. . 3 ⊢ ∘func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩)
2	1	a1i 11	. 2 ⊢ (𝜑 → ∘func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩))
3		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
4	3	fveq2d 6882	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1st ‘𝑔) = (1st ‘𝐺))
5		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
6	5	fveq2d 6882	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (1st ‘𝑓) = (1st ‘𝐹))
7	4, 6	coeq12d 5844	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ((1st ‘𝑔) ∘ (1st ‘𝑓)) = ((1st ‘𝐺) ∘ (1st ‘𝐹)))
8	5	fveq2d 6882	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘𝑓) = (2nd ‘𝐹))
9	8	dmeqd 5889	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → dom (2nd ‘𝑓) = dom (2nd ‘𝐹))
10		cofuval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐶)
11		relfunc 17951	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
12		cofuval.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
13		1st2ndbr 8039	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
14	11, 12, 13	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
15	10, 14	funcfn2 17958	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘𝐹) Fn (𝐵 × 𝐵))
16	15	fndmd 6637	. . . . . . . 8 ⊢ (𝜑 → dom (2nd ‘𝐹) = (𝐵 × 𝐵))
17	16	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → dom (2nd ‘𝐹) = (𝐵 × 𝐵))
18	9, 17	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → dom (2nd ‘𝑓) = (𝐵 × 𝐵))
19	18	dmeqd 5889	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → dom dom (2nd ‘𝑓) = dom (𝐵 × 𝐵))
20		dmxpid 5914	. . . . 5 ⊢ dom (𝐵 × 𝐵) = 𝐵
21	19, 20	eqtrdi 2811	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → dom dom (2nd ‘𝑓) = 𝐵)
22	3	fveq2d 6882	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘𝑔) = (2nd ‘𝐺))
23	6	fveq1d 6880	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ((1st ‘𝑓)‘𝑥) = ((1st ‘𝐹)‘𝑥))
24	6	fveq1d 6880	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ((1st ‘𝑓)‘𝑦) = ((1st ‘𝐹)‘𝑦))
25	22, 23, 24	oveq123d 7434	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) = (((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)))
26	8	oveqd 7430	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑥(2nd ‘𝑓)𝑦) = (𝑥(2nd ‘𝐹)𝑦))
27	25, 26	coeq12d 5844	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)) = ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))
28	21, 21, 27	mpoeq123dv 7488	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))))
29	7, 28	opeq12d 4841	. 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩ = ⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩)
30		cofuval.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
31	30	elexd 3473	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
32	12	elexd 3473	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
33		opex 5439	. . 3 ⊢ ⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩ ∈ V
34	33	a1i 11	. 2 ⊢ (𝜑 → ⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩ ∈ V)
35	2, 29, 31, 32, 34	ovmpod 7565	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = ⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ⟨cop 4590   class class class wbr 5103   × cxp 5653  dom cdm 5655   ∘ ccom 5659  Rel wrel 5660  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  1^st c1st 7984  2^nd c2nd 7985  Basecbs 17301   Func cfunc 17943   ∘_func ccofu 17945

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-map 8828  df-ixp 8905  df-func 17947  df-cofu 17949

This theorem is used by:  cofu1st  17972  cofu2nd  17974  cofuval2  17976  cofucl  17977  cofuass  17978  cofulid  17979  cofurid  17980  prf1st  18292  prf2nd  18293  cofidvala  50042  cofuoppf  50076