Theorem cofidvala 49591

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 17849	. . 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 49551	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
8	7	funcrcl2 49554	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
9		cofidvala.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐷)
10	6, 2, 8, 9	idfuval 17843	. . 3 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
11	1, 5, 10	3eqtr3d 2779	. 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 6854	. . . 4 ⊢ 𝐵 ∈ V
13		resiexg 7863	. . . 4 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
14	12, 13	ax-mp 5	. . 3 ⊢ ( I ↾ 𝐵) ∈ V
15	12, 12	xpex 7707	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
16	15	mptex 7178	. . 3 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) ∈ V
17	14, 16	opth2 5433	. 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 1542   ∈ wcel 2114  Vcvv 3429  ⟨cop 4573   ↦ cmpt 5166   I cid 5525   × cxp 5629   ↾ cres 5633   ∘ ccom 5635  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1^st c1st 7940  2^nd c2nd 7941  Basecbs 17179  Hom chom 17231   Func cfunc 17821  id_funccidfu 17822   ∘_func ccofu 17823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-ixp 8846  df-func 17825  df-idfu 17826  df-cofu 17827

This theorem is referenced by:  cofidf1a  49593