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Theorem invffval 17700
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 6887 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 invfval.b . . . . . 6 𝐵 = (Base‘𝐶)
53, 4eqtr4di 2791 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fveq2 6887 . . . . . . . 8 (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶))
7 invfval.s . . . . . . . 8 𝑆 = (Sect‘𝐶)
86, 7eqtr4di 2791 . . . . . . 7 (𝑐 = 𝐶 → (Sect‘𝑐) = 𝑆)
98oveqd 7420 . . . . . 6 (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥𝑆𝑦))
108oveqd 7420 . . . . . . 7 (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦𝑆𝑥))
1110cnveqd 5872 . . . . . 6 (𝑐 = 𝐶(𝑦(Sect‘𝑐)𝑥) = (𝑦𝑆𝑥))
129, 11ineq12d 4211 . . . . 5 (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥)) = ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥)))
135, 5, 12mpoeq123dv 7478 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))) = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
14 df-inv 17690 . . . 4 Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))))
154fvexi 6901 . . . . 5 𝐵 ∈ V
1615, 15mpoex 8060 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))) ∈ V
1713, 14, 16fvmpt 6993 . . 3 (𝐶 ∈ Cat → (Inv‘𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
182, 17syl 17 . 2 (𝜑 → (Inv‘𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
191, 18eqtrid 2785 1 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cin 3945  ccnv 5673  cfv 6539  (class class class)co 7403  cmpo 7405  Basecbs 17139  Catccat 17603  Sectcsect 17686  Invcinv 17687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7969  df-2nd 7970  df-inv 17690
This theorem is referenced by:  invfval  17701  isoval  17707
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