Theorem invfval 17388

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
invfval	⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)))

Proof of Theorem invfval

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

Step	Hyp	Ref	Expression
1		invfval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		invfval.n	. . 3 ⊢ 𝑁 = (Inv‘𝐶)
3		invfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		invfval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		invfval.s	. . 3 ⊢ 𝑆 = (Sect‘𝐶)
6	1, 2, 3, 4, 4, 5	invffval 17387	. 2 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))
7		simprl 767	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
8		simprr 769	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
9	7, 8	oveq12d 7273	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝑆𝑦) = (𝑋𝑆𝑌))
10	8, 7	oveq12d 7273	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋))
11	10	cnveqd 5773	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ◡(𝑦𝑆𝑥) = ◡(𝑌𝑆𝑋))
12	9, 11	ineq12d 4144	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)))
13		invfval.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14		ovex 7288	. . . 4 ⊢ (𝑋𝑆𝑌) ∈ V
15	14	inex1 5236	. . 3 ⊢ ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)) ∈ V
16	15	a1i 11	. 2 ⊢ (𝜑 → ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)) ∈ V)
17	6, 12, 4, 13, 16	ovmpod 7403	1 ⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882  ^◡ccnv 5579  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Catccat 17290  Sectcsect 17373  Invcinv 17374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-inv 17377

This theorem is referenced by:  isinv  17389  invss  17390  dfiso2  17401  oppcinv  17409