Theorem fuco11bALT 50791

Description: Alternate proof of fuco11b 50790. (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 7454	. 2 ⊢ (𝐺𝑂𝐹) = (𝑂‘⟨𝐺, 𝐹⟩)
2		relfunc 18087	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
3		fuco11b.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
4		1st2nd 8084	. . . . 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 18087	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
7		fuco11b.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
8		1st2nd 8084	. . . . 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 7469	. . 3 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
11		1st2ndbr 8087	. . . . . . . 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 50524	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
14		1st2ndbr 8087	. . . . . . . 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 50524	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
17	15	funcrcl3 50525	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
18		eqidd 2764	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (⟨𝐶, 𝐷⟩ ∘F 𝐸))
19	13, 16, 17, 18	fucoelvv 50773	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V))
20		1st2nd2 8074	. . . . 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 4853	. . . 4 ⊢ (𝜑 → ⟨𝐺, 𝐹⟩ = ⟨⟨(1st ‘𝐺), (2nd ‘𝐺)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
23	21, 12, 15, 22	fuco11 50779	. . 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 6918	. . 3 ⊢ (𝜑 → ((1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))‘⟨𝐺, 𝐹⟩) = (𝑂‘⟨𝐺, 𝐹⟩))
26	10, 23, 25	3eqtr2rd 2805	. 2 ⊢ (𝜑 → (𝑂‘⟨𝐺, 𝐹⟩) = (𝐺 ∘func 𝐹))
27	1, 26	eqtrid 2810	1 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺 ∘func 𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4840   class class class wbr 5108   × cxp 5656  Rel wrel 5663  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033  2^nd c2nd 8034  Catccat 17888   Func cfunc 18079   ∘_func ccofu 18081   ∘_F cfuco 50769

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-func 18083  df-cofu 18085  df-fuco 50770

This theorem is used by: (None)