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Theorem invfval 17807
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 17806 . 2 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
7 simprl 771 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
8 simprr 773 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
97, 8oveq12d 7449 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝑆𝑦) = (𝑋𝑆𝑌))
108, 7oveq12d 7449 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋))
1110cnveqd 5889 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋))
129, 11ineq12d 4229 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥)) = ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)))
13 invfval.y . 2 (𝜑𝑌𝐵)
14 ovex 7464 . . . 4 (𝑋𝑆𝑌) ∈ V
1514inex1 5323 . . 3 ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)) ∈ V
1615a1i 11 . 2 (𝜑 → ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)) ∈ V)
176, 12, 4, 13, 16ovmpod 7585 1 (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  ccnv 5688  cfv 6563  (class class class)co 7431  Basecbs 17245  Catccat 17709  Sectcsect 17792  Invcinv 17793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-inv 17796
This theorem is referenced by:  isinv  17808  invss  17809  dfiso2  17820  oppcinv  17828
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