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Theorem invffval 17701

Description: Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)

**Hypotheses**
Ref	Expression
invfval.b	⊢ 𝐵 = (Base‘𝐶)
invfval.n	⊢ 𝑁 = (Inv‘𝐶)
invfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
invfval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invfval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
invfval.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,𝑋,𝑦 𝑥,𝑌,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 6888	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4		invfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	eqtr4di 2790	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6		fveq2 6888	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶))
7		invfval.s	. . . . . . . 8 ⊢ 𝑆 = (Sect‘𝐶)
8	6, 7	eqtr4di 2790	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Sect‘𝑐) = 𝑆)
9	8	oveqd 7422	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥𝑆𝑦))
10	8	oveqd 7422	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦𝑆𝑥))
11	10	cnveqd 5873	. . . . . 6 ⊢ (𝑐 = 𝐶 → ^◡(𝑦(Sect‘𝑐)𝑥) = ^◡(𝑦𝑆𝑥))
12	9, 11	ineq12d 4212	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ ^◡(𝑦(Sect‘𝑐)𝑥)) = ((𝑥𝑆𝑦) ∩ ^◡(𝑦𝑆𝑥)))
13	5, 5, 12	mpoeq123dv 7480	. . . 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 17691	. . . 4 ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ^◡(𝑦(Sect‘𝑐)𝑥))))
15	4	fvexi 6902	. . . . 5 ⊢ 𝐵 ∈ V
16	15, 15	mpoex 8062	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ^◡(𝑦𝑆𝑥))) ∈ V
17	13, 14, 16	fvmpt 6995	. . 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 2784	1 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ^◡(𝑦𝑆𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∩ cin 3946 ^◡ccnv 5674 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 Basecbs 17140 Catccat 17604 Sectcsect 17687 Invcinv 17688

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-inv 17691

This theorem is referenced by: invfval 17702 isoval 17708

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