Theorem invfn 48992

Description: The function value of the function returning the inverses of a category is a function over the Cartesian square of the base set of the category. Simplifies isofn 17713 (see isofnALT 48993). (Contributed by Zhi Wang, 27-Oct-2025.)

Assertion

Ref	Expression
invfn	⊢ (𝐶 ∈ Cat → (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))

Proof of Theorem invfn

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

Step	Hyp	Ref	Expression
1		ovex 7402	. . . . . 6 ⊢ (𝑥(Sect‘𝐶)𝑦) ∈ V
2	1	inex1 5267	. . . . 5 ⊢ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)) ∈ V
3	2	a1i 11	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)) ∈ V)
4	3	ralrimivva 3178	. . 3 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)) ∈ V)
5		eqid 2729	. . . 4 ⊢ (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)))
6	5	fnmpo 8027	. . 3 ⊢ (∀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)))
7	4, 6	syl 17	. 2 ⊢ (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) Fn ((Base‘𝐶) × (Base‘𝐶)))
8		df-inv 17686	. . . 4 ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))))
9		fveq2 6840	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
10		fveq2 6840	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶))
11	10	oveqd 7386	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥(Sect‘𝐶)𝑦))
12	10	oveqd 7386	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦(Sect‘𝐶)𝑥))
13	12	cnveqd 5829	. . . . . 6 ⊢ (𝑐 = 𝐶 → ◡(𝑦(Sect‘𝑐)𝑥) = ◡(𝑦(Sect‘𝐶)𝑥))
14	11, 13	ineq12d 4180	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)) = ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥)))
15	9, 9, 14	mpoeq123dv 7444	. . . 4 ⊢ (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))))
16		id 22	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
17		fvex 6853	. . . . . 6 ⊢ (Base‘𝐶) ∈ V
18	17, 17	pm3.2i 470	. . . . 5 ⊢ ((Base‘𝐶) ∈ V ∧ (Base‘𝐶) ∈ V)
19		mpoexga 8035	. . . . 5 ⊢ (((Base‘𝐶) ∈ V ∧ (Base‘𝐶) ∈ V) → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) ∈ V)
20	18, 19	mp1i 13	. . . 4 ⊢ (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))) ∈ V)
21	8, 15, 16, 20	fvmptd3 6973	. . 3 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ ◡(𝑦(Sect‘𝐶)𝑥))))
22	21	fneq1d 6593	. 2 ⊢ (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))))
23	7, 22	mpbird 257	1 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ∩ cin 3910   × cxp 5629  ^◡ccnv 5630   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  Basecbs 17155  Catccat 17601  Sectcsect 17682  Invcinv 17683

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  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 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-inv 17686

This theorem is referenced by:  isofnALT  48993  invpropdlem  49000