Theorem fuco11bALT 49083

Description: Alternate proof of fuco11b 49082. (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 17879	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
3		fuco11b.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
4		1st2nd 8046	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐺 ∈ (𝐷 Func 𝐸)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
5	2, 3, 4	sylancr 587	. . . 4 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
6		relfunc 17879	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
7		fuco11b.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
8		1st2nd 8046	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
9	6, 7, 8	sylancr 587	. . . 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 8049	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
12	6, 7, 11	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
13	12	funcrcl2 48937	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
14		1st2ndbr 8049	. . . . . . . 8 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐺 ∈ (𝐷 Func 𝐸)) → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
15	2, 3, 14	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
16	15	funcrcl2 48937	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
17	15	funcrcl3 48938	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
18		eqidd 2735	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (⟨𝐶, 𝐷⟩ ∘F 𝐸))
19	13, 16, 17, 18	fucoelvv 49065	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V))
20		1st2nd2 8035	. . . . 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 17	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)), (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟩)
22	5, 9	opeq12d 4861	. . . 4 ⊢ (𝜑 → ⟨𝐺, 𝐹⟩ = ⟨⟨(1st ‘𝐺), (2nd ‘𝐺)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
23	21, 12, 15, 22	fuco11 49071	. . 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 6888	. . 3 ⊢ (𝜑 → ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝐺, 𝐹⟩) = (𝑂‘⟨𝐺, 𝐹⟩))
26	10, 23, 25	3eqtr2rd 2776	. 2 ⊢ (𝜑 → (𝑂‘⟨𝐺, 𝐹⟩) = (𝐺 ∘func 𝐹))
27	1, 26	eqtrid 2781	1 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺 ∘func 𝐹))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  Vcvv 3463  ⟨cop 4612   class class class wbr 5123   × cxp 5663  Rel wrel 5670  ‘cfv 6541  (class class class)co 7413  1^st c1st 7994  2^nd c2nd 7995  Catccat 17679   Func cfunc 17871   ∘_func ccofu 17873   ∘_F cfuco 49061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997  df-func 17875  df-cofu 17877  df-fuco 49062

This theorem is referenced by: (None)