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Theorem invffval 17802
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 6906 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 invfval.b . . . . . 6 𝐵 = (Base‘𝐶)
53, 4eqtr4di 2795 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fveq2 6906 . . . . . . . 8 (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶))
7 invfval.s . . . . . . . 8 𝑆 = (Sect‘𝐶)
86, 7eqtr4di 2795 . . . . . . 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 5886 . . . . . 6 (𝑐 = 𝐶(𝑦(Sect‘𝑐)𝑥) = (𝑦𝑆𝑥))
129, 11ineq12d 4221 . . . . 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 17792 . . . 4 Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))))
154fvexi 6920 . . . . 5 𝐵 ∈ V
1615, 15mpoex 8104 . . . 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 2789 1 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cin 3950  ccnv 5684  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  Catccat 17707  Sectcsect 17788  Invcinv 17789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-inv 17792
This theorem is referenced by:  invfval  17803  isoval  17809
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