Theorem cofidval 49748

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

Hypotheses

Ref	Expression
cofidval.i	⊢ 𝐼 = (idfunc‘𝐷)
cofidval.b	⊢ 𝐵 = (Base‘𝐷)
cofidval.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
cofidval.k	⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)
cofidval.o	⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)
cofidval.h	⊢ 𝐻 = (Hom ‘𝐷)

Assertion

Ref	Expression
cofidval	⊢ (𝜑 → ((𝐾 ∘ 𝐹) = ( I ↾ 𝐵) ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑧,𝐵   𝑧,𝐷   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑧,𝐻   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   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)   𝐾(𝑧)   𝐿(𝑧)

Proof of Theorem cofidval

Step	Hyp	Ref	Expression
1		cofidval.o	. . 3 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)
2		cofidval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
3		cofidval.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
4		cofidval.k	. . . 4 ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)
5	2, 3, 4	cofuval2 17934	. . 3 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨(𝐾 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
6		cofidval.i	. . . 4 ⊢ 𝐼 = (idfunc‘𝐷)
7	3	funcrcl2 49708	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		cofidval.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐷)
9	6, 2, 7, 8	idfuval 17923	. . 3 ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
10	1, 5, 9	3eqtr3d 2808	. 2 ⊢ (𝜑 → ⟨(𝐾 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩ = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
11	2	fvexi 6885	. . . 4 ⊢ 𝐵 ∈ V
12		resiexg 7897	. . . 4 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
13	11, 12	ax-mp 5	. . 3 ⊢ ( I ↾ 𝐵) ∈ V
14	11, 11	xpex 7740	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
15	14	mptex 7211	. . 3 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))) ∈ V
16	13, 15	opth2 5453	. 2 ⊢ (⟨(𝐾 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩ = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩ ↔ ((𝐾 ∘ 𝐹) = ( I ↾ 𝐵) ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))))
17	10, 16	sylib 221	1 ⊢ (𝜑 → ((𝐾 ∘ 𝐹) = ( I ↾ 𝐵) ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ⟨cop 4591   class class class wbr 5105   ↦ cmpt 5186   I cid 5546   × cxp 5650   ↾ cres 5654   ∘ ccom 5656  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  Basecbs 17259  Hom chom 17311   Func cfunc 17901  id_funccidfu 17902   ∘_func ccofu 17903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-ixp 8884  df-func 17905  df-idfu 17906  df-cofu 17907

This theorem is referenced by:  cofidf1  49750