Theorem cofidvala 49105

Description: The property "𝐹 is a section of 𝐺 " in a category of small categories (in a universe); expressed explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)

Hypotheses

Ref	Expression
cofidvala.i	⊢ 𝐼 = (idfunc‘𝐷)
cofidvala.b	⊢ 𝐵 = (Base‘𝐷)
cofidvala.f	⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
cofidvala.g	⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))
cofidvala.o	⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)
cofidvala.h	⊢ 𝐻 = (Hom ‘𝐷)

Assertion

Ref	Expression
cofidvala	⊢ (𝜑 → (((1st ‘𝐺) ∘ (1st ‘𝐹)) = ( I ↾ 𝐵) ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑧,𝐵   𝑧,𝐷   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑧,𝐻   𝜑,𝑥,𝑦   𝜑,𝑧

Allowed substitution hints:   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦,𝑧)   𝐹(𝑧)   𝐺(𝑧)   𝐻(𝑥,𝑦)   𝐼(𝑥,𝑦,𝑧)

Proof of Theorem cofidvala

Step	Hyp	Ref	Expression
1		cofidvala.o	. . 3 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)
2		cofidvala.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
3		cofidvala.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
4		cofidvala.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))
5	2, 3, 4	cofuval 17844	. . 3 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = ⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩)
6		cofidvala.i	. . . 4 ⊢ 𝐼 = (idfunc‘𝐷)
7	3	func1st2nd 49065	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
8	7	funcrcl2 49068	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
9		cofidvala.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐷)
10	6, 2, 8, 9	idfuval 17838	. . 3 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
11	1, 5, 10	3eqtr3d 2772	. 2 ⊢ (𝜑 → ⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩ = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
12	2	fvexi 6872	. . . 4 ⊢ 𝐵 ∈ V
13		resiexg 7888	. . . 4 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
14	12, 13	ax-mp 5	. . 3 ⊢ ( I ↾ 𝐵) ∈ V
15	12, 12	xpex 7729	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
16	15	mptex 7197	. . 3 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) ∈ V
17	14, 16	opth2 5440	. 2 ⊢ (⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩ = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩ ↔ (((1st ‘𝐺) ∘ (1st ‘𝐹)) = ( I ↾ 𝐵) ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))))
18	11, 17	sylib 218	1 ⊢ (𝜑 → (((1st ‘𝐺) ∘ (1st ‘𝐹)) = ( I ↾ 𝐵) ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⟨cop 4595   ↦ cmpt 5188   I cid 5532   × cxp 5636   ↾ cres 5640   ∘ ccom 5642  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179  Hom chom 17231   Func cfunc 17816  id_funccidfu 17817   ∘_func ccofu 17818

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801  df-ixp 8871  df-func 17820  df-idfu 17821  df-cofu 17822

This theorem is referenced by:  cofidf1a  49107