Theorem fuco11bALT 49300

Description: Alternate proof of fuco11b 49299. (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 7372	. 2 ⊢ (𝐺𝑂𝐹) = (𝑂‘⟨𝐺, 𝐹⟩)
2		relfunc 17800	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
3		fuco11b.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
4		1st2nd 7997	. . . . 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 17800	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
7		fuco11b.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
8		1st2nd 7997	. . . . 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 7387	. . 3 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
11		1st2ndbr 8000	. . . . . . . 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 49041	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
14		1st2ndbr 8000	. . . . . . . 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 49041	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
17	15	funcrcl3 49042	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
18		eqidd 2730	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (⟨𝐶, 𝐷⟩ ∘F 𝐸))
19	13, 16, 17, 18	fucoelvv 49282	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V))
20		1st2nd2 7986	. . . . 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 4841	. . . 4 ⊢ (𝜑 → ⟨𝐺, 𝐹⟩ = ⟨⟨(1st ‘𝐺), (2nd ‘𝐺)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
23	21, 12, 15, 22	fuco11 49288	. . 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 6842	. . 3 ⊢ (𝜑 → ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝐺, 𝐹⟩) = (𝑂‘⟨𝐺, 𝐹⟩))
26	10, 23, 25	3eqtr2rd 2771	. 2 ⊢ (𝜑 → (𝑂‘⟨𝐺, 𝐹⟩) = (𝐺 ∘func 𝐹))
27	1, 26	eqtrid 2776	1 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺 ∘func 𝐹))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ⟨cop 4591   class class class wbr 5102   × cxp 5629  Rel wrel 5636  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  Catccat 17601   Func cfunc 17792   ∘_func ccofu 17794   ∘_F cfuco 49278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-func 17796  df-cofu 17798  df-fuco 49279

This theorem is referenced by: (None)