Theorem fuco11bALT 50264

Description: Alternate proof of fuco11b 50263. (Contributed by Zhi Wang, 11-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
fuco11b.o	⊢ (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = 𝑂)
fuco11b.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
fuco11b.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))

Assertion

Ref	Expression
fuco11bALT	⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺 ∘func 𝐹))

Proof of Theorem fuco11bALT

Step	Hyp	Ref	Expression
1		df-ov 7416	. 2 ⊢ (𝐺𝑂𝐹) = (𝑂‘⟨𝐺, 𝐹⟩)
2		relfunc 17951	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
3		fuco11b.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
4		1st2nd 8036	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐺 ∈ (𝐷 Func 𝐸)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
5	2, 3, 4	sylancr 599	. . . 4 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
6		relfunc 17951	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
7		fuco11b.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
8		1st2nd 8036	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
9	6, 7, 8	sylancr 599	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
10	5, 9	oveq12d 7431	. . 3 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
11		1st2ndbr 8039	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
12	6, 7, 11	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
13	12	funcrcl2 50005	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
14		1st2ndbr 8039	. . . . . . . 8 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐺 ∈ (𝐷 Func 𝐸)) → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
15	2, 3, 14	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
16	15	funcrcl2 50005	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
17	15	funcrcl3 50006	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
18		eqidd 2761	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (⟨𝐶, 𝐷⟩ ∘F 𝐸))
19	13, 16, 17, 18	fucoelvv 50246	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V))
20		1st2nd2 8025	. . . . 5 ⊢ ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V) → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)), (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟩)
21	19, 20	syl 18	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)), (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟩)
22	5, 9	opeq12d 4841	. . . 4 ⊢ (𝜑 → ⟨𝐺, 𝐹⟩ = ⟨⟨(1st ‘𝐺), (2nd ‘𝐺)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
23	21, 12, 15, 22	fuco11 50252	. . 3 ⊢ (𝜑 → ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝐺, 𝐹⟩) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
24		fuco11b.o	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = 𝑂)
25	24	fveq1d 6880	. . 3 ⊢ (𝜑 → ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝐺, 𝐹⟩) = (𝑂‘⟨𝐺, 𝐹⟩))
26	10, 23, 25	3eqtr2rd 2802	. 2 ⊢ (𝜑 → (𝑂‘⟨𝐺, 𝐹⟩) = (𝐺 ∘func 𝐹))
27	1, 26	eqtrid 2807	1 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺 ∘func 𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ⟨cop 4590   class class class wbr 5103   × cxp 5653  Rel wrel 5660  ‘cfv 6533  (class class class)co 7413  1^st c1st 7984  2^nd c2nd 7985  Catccat 17752   Func cfunc 17943   ∘_func ccofu 17945   ∘_F cfuco 50242

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-reu 3366  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-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-func 17947  df-cofu 17949  df-fuco 50243

This theorem is used by: (None)