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Theorem invfval 16242
 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 (𝜑𝐶 ∈ Cat)
invfval.x (𝜑𝑋𝐵)
invfval.y (𝜑𝑌𝐵)
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.
StepHypRef Expression
1 invfval.b . . 3 𝐵 = (Base‘𝐶)
2 invfval.n . . 3 𝑁 = (Inv‘𝐶)
3 invfval.c . . 3 (𝜑𝐶 ∈ Cat)
4 invfval.x . . 3 (𝜑𝑋𝐵)
5 invfval.s . . 3 𝑆 = (Sect‘𝐶)
61, 2, 3, 4, 4, 5invffval 16241 . 2 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
7 simprl 790 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
8 simprr 792 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
97, 8oveq12d 6567 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝑆𝑦) = (𝑋𝑆𝑌))
108, 7oveq12d 6567 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋))
1110cnveqd 5220 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋))
129, 11ineq12d 3777 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥)) = ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)))
13 invfval.y . 2 (𝜑𝑌𝐵)
14 ovex 6577 . . . 4 (𝑋𝑆𝑌) ∈ V
1514inex1 4727 . . 3 ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)) ∈ V
1615a1i 11 . 2 (𝜑 → ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)) ∈ V)
176, 12, 4, 13, 16ovmpt2d 6686 1 (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ◡ccnv 5037  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Catccat 16148  Sectcsect 16227  Invcinv 16228 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-inv 16231 This theorem is referenced by:  isinv  16243  invss  16244  dfiso2  16255  oppcinv  16263
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