Theorem invffval 17683

Description: Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) Removed redundant hypotheses. (Revised by Zhi Wang, 27-Oct-2025.)

Hypotheses

Ref	Expression
invfval.b	⊢ 𝐵 = (Base‘𝐶)
invfval.n	⊢ 𝑁 = (Inv‘𝐶)
invfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
invffval.s	⊢ 𝑆 = (Sect‘𝐶)

Assertion

Ref	Expression
invffval	⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))

Distinct variable groups:   𝑥,𝑦,𝐵   𝜑,𝑥,𝑦   𝑥,𝐶,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝑁(𝑥,𝑦)

Proof of Theorem invffval

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		invfval.n	. 2 ⊢ 𝑁 = (Inv‘𝐶)
2		invfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		fveq2 6826	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4		invfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	eqtr4di 2782	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6		fveq2 6826	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶))
7		invffval.s	. . . . . . . 8 ⊢ 𝑆 = (Sect‘𝐶)
8	6, 7	eqtr4di 2782	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = 𝑆)
9	8	oveqd 7370	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥𝑆𝑦))
10	8	oveqd 7370	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦𝑆𝑥))
11	10	cnveqd 5822	. . . . . 6 ⊢ (𝑐 = 𝐶 → ◡(𝑦(Sect‘𝑐)𝑥) = ◡(𝑦𝑆𝑥))
12	9, 11	ineq12d 4174	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)) = ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)))
13	5, 5, 12	mpoeq123dv 7428	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))
14		df-inv 17673	. . . 4 ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))))
15	4	fvexi 6840	. . . . 5 ⊢ 𝐵 ∈ V
16	15, 15	mpoex 8021	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))) ∈ V
17	13, 14, 16	fvmpt 6934	. . 3 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))
18	2, 17	syl 17	. 2 ⊢ (𝜑 → (Inv‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))
19	1, 18	eqtrid 2776	1 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∩ cin 3904  ^◡ccnv 5622  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  Basecbs 17138  Catccat 17588  Sectcsect 17669  Invcinv 17670

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-inv 17673

This theorem is referenced by:  invfval  17684  isoval  17690  invrcl2  49011  invpropdlem  49024