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Theorem invffval 17806
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
invffval (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((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.
StepHypRef Expression
1 invfval.n . 2 𝑁 = (Inv‘𝐶)
2 invfval.c . . 3 (𝜑𝐶 ∈ Cat)
3 fveq2 6907 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 invfval.b . . . . . 6 𝐵 = (Base‘𝐶)
53, 4eqtr4di 2793 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fveq2 6907 . . . . . . . 8 (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶))
7 invfval.s . . . . . . . 8 𝑆 = (Sect‘𝐶)
86, 7eqtr4di 2793 . . . . . . 7 (𝑐 = 𝐶 → (Sect‘𝑐) = 𝑆)
98oveqd 7448 . . . . . 6 (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥𝑆𝑦))
108oveqd 7448 . . . . . . 7 (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦𝑆𝑥))
1110cnveqd 5889 . . . . . 6 (𝑐 = 𝐶(𝑦(Sect‘𝑐)𝑥) = (𝑦𝑆𝑥))
129, 11ineq12d 4229 . . . . 5 (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥)) = ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥)))
135, 5, 12mpoeq123dv 7508 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))) = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
14 df-inv 17796 . . . 4 Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))))
154fvexi 6921 . . . . 5 𝐵 ∈ V
1615, 15mpoex 8103 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))) ∈ V
1713, 14, 16fvmpt 7016 . . 3 (𝐶 ∈ Cat → (Inv‘𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
182, 17syl 17 . 2 (𝜑 → (Inv‘𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
191, 18eqtrid 2787 1 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cin 3962  ccnv 5688  cfv 6563  (class class class)co 7431  cmpo 7433  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:  invfval  17807  isoval  17813
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