Theorem isofn 17711

Description: The function value of the function returning the isomorphisms of a category is a function over the Cartesian square 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 7853	. . . . . 6 ⊢ (𝑥 ∈ V → dom 𝑥 ∈ V)
2	1	adantl 481	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ V) → dom 𝑥 ∈ V)
3	2	ralrimiva 3130	. . . 4 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ V dom 𝑥 ∈ V)
4		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ V ↦ dom 𝑥) = (𝑥 ∈ V ↦ dom 𝑥)
5	4	fnmpt 6640	. . . 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 7401	. . . . . . . 8 ⊢ (𝑥(Sect‘𝐶)𝑦) ∈ V
8	7	inex1 5264	. . . . . . 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 3181	. . . . 5 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)) ∈ V)
11		eqid 2737	. . . . . 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 8023	. . . . 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 17684	. . . . . 6 ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))))
15		fveq2 6842	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
16		fveq2 6842	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶))
17	16	oveqd 7385	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥(Sect‘𝐶)𝑦))
18	16	oveqd 7385	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦(Sect‘𝐶)𝑥))
19	18	cnveqd 5832	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → ◡(𝑦(Sect‘𝑐)𝑥) = ◡(𝑦(Sect‘𝐶)𝑥))
20	17, 19	ineq12d 4175	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)) = ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))
21	15, 15, 20	mpoeq123dv 7443	. . . . . 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 6855	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
24	23, 23	pm3.2i 470	. . . . . . 7 ⊢ ((Base‘𝐶) ∈ V ∧ (Base‘𝐶) ∈ V)
25		mpoexga 8031	. . . . . . 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 6973	. . . . 5 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
28	27	fneq1d 6593	. . . 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 3960	. . . 4 ⊢ ran (Inv‘𝐶) ⊆ V
31	30	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → ran (Inv‘𝐶) ⊆ V)
32		fnco 6618	. . 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 1374	. 2 ⊢ (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) Fn ((Base‘𝐶) × (Base‘𝐶)))
34		isofval 17693	. . 3 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
35	34	fneq1d 6593	. 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 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903   ↦ cmpt 5181   × cxp 5630  ^◡ccnv 5631  dom cdm 5632  ran crn 5633   ∘ ccom 5636   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  Basecbs 17148  Catccat 17599  Sectcsect 17680  Invcinv 17681  Isociso 17682

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-inv 17684  df-iso 17685

This theorem is referenced by:  brcic  17734  ciclcl  17738  cicrcl  17739  cicer  17742  isofval2  49385  isopropdlem  49393  relcic  49398  upeu4  49549