Theorem isofn 17719

Description: The function value of the function returning the isomorphisms of a category is a function over the square product of the base set of the category. (Contributed by AV, 5-Apr-2020.)

Assertion

Ref	Expression
isofn	⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))

Proof of Theorem isofn

Dummy variables 𝑐 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmexg 7891	. . . . . 6 ⊢ (𝑥 ∈ V → dom 𝑥 ∈ V)
2	1	adantl 483	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ V) → dom 𝑥 ∈ V)
3	2	ralrimiva 3147	. . . 4 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ V dom 𝑥 ∈ V)
4		eqid 2733	. . . . 5 ⊢ (𝑥 ∈ V ↦ dom 𝑥) = (𝑥 ∈ V ↦ dom 𝑥)
5	4	fnmpt 6688	. . . 4 ⊢ (∀𝑥 ∈ V dom 𝑥 ∈ V → (𝑥 ∈ V ↦ dom 𝑥) Fn V)
6	3, 5	syl 17	. . 3 ⊢ (𝐶 ∈ Cat → (𝑥 ∈ V ↦ dom 𝑥) Fn V)
7		ovex 7439	. . . . . . . 8 ⊢ (𝑥(Sect‘𝐶)𝑦) ∈ V
8	7	inex1 5317	. . . . . . 7 ⊢ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)) ∈ V
9	8	a1i 11	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)) ∈ V)
10	9	ralrimivva 3201	. . . . 5 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)) ∈ V)
11		eqid 2733	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))
12	11	fnmpo 8052	. . . . 5 ⊢ (∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)) ∈ V → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) Fn ((Base‘𝐶) × (Base‘𝐶)))
13	10, 12	syl 17	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) Fn ((Base‘𝐶) × (Base‘𝐶)))
14		df-inv 17692	. . . . . 6 ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))))
15		fveq2 6889	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
16		fveq2 6889	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶))
17	16	oveqd 7423	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥(Sect‘𝐶)𝑦))
18	16	oveqd 7423	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦(Sect‘𝐶)𝑥))
19	18	cnveqd 5874	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → ◡(𝑦(Sect‘𝑐)𝑥) = ◡(𝑦(Sect‘𝐶)𝑥))
20	17, 19	ineq12d 4213	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)) = ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))
21	15, 15, 20	mpoeq123dv 7481	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
22		id 22	. . . . . 6 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
23		fvex 6902	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
24	23, 23	pm3.2i 472	. . . . . . 7 ⊢ ((Base‘𝐶) ∈ V ∧ (Base‘𝐶) ∈ V)
25		mpoexga 8061	. . . . . . 7 ⊢ (((Base‘𝐶) ∈ V ∧ (Base‘𝐶) ∈ V) → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) ∈ V)
26	24, 25	mp1i 13	. . . . . 6 ⊢ (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) ∈ V)
27	14, 21, 22, 26	fvmptd3 7019	. . . . 5 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
28	27	fneq1d 6640	. . . 4 ⊢ (𝐶 ∈ Cat → ((Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) Fn ((Base‘𝐶) × (Base‘𝐶))))
29	13, 28	mpbird 257	. . 3 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
30		ssv 4006	. . . 4 ⊢ ran (Inv‘𝐶) ⊆ V
31	30	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → ran (Inv‘𝐶) ⊆ V)
32		fnco 6665	. . 3 ⊢ (((𝑥 ∈ V ↦ dom 𝑥) Fn V ∧ (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ran (Inv‘𝐶) ⊆ V) → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) Fn ((Base‘𝐶) × (Base‘𝐶)))
33	6, 29, 31, 32	syl3anc 1372	. 2 ⊢ (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) Fn ((Base‘𝐶) × (Base‘𝐶)))
34		isofval 17701	. . 3 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
35	34	fneq1d 6640	. 2 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) Fn ((Base‘𝐶) × (Base‘𝐶))))
36	33, 35	mpbird 257	1 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ∩ cin 3947   ⊆ wss 3948   ↦ cmpt 5231   × cxp 5674  ^◡ccnv 5675  dom cdm 5676  ran crn 5677   ∘ ccom 5680   Fn wfn 6536  ‘cfv 6541  (class class class)co 7406   ∈ cmpo 7408  Basecbs 17141  Catccat 17605  Sectcsect 17688  Invcinv 17689  Isociso 17690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-inv 17692  df-iso 17693

This theorem is referenced by:  brcic  17742  ciclcl  17746  cicrcl  17747  cicer  17750