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Theorem invffval 16639
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 6353 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 invfval.b . . . . . 6 𝐵 = (Base‘𝐶)
53, 4syl6eqr 2812 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fveq2 6353 . . . . . . . 8 (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶))
7 invfval.s . . . . . . . 8 𝑆 = (Sect‘𝐶)
86, 7syl6eqr 2812 . . . . . . 7 (𝑐 = 𝐶 → (Sect‘𝑐) = 𝑆)
98oveqd 6831 . . . . . 6 (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥𝑆𝑦))
108oveqd 6831 . . . . . . 7 (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦𝑆𝑥))
1110cnveqd 5453 . . . . . 6 (𝑐 = 𝐶(𝑦(Sect‘𝑐)𝑥) = (𝑦𝑆𝑥))
129, 11ineq12d 3958 . . . . 5 (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥)) = ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥)))
135, 5, 12mpt2eq123dv 6883 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))) = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
14 df-inv 16629 . . . 4 Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))))
15 fvex 6363 . . . . . 6 (Base‘𝐶) ∈ V
164, 15eqeltri 2835 . . . . 5 𝐵 ∈ V
1716, 16mpt2ex 7416 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))) ∈ V
1813, 14, 17fvmpt 6445 . . 3 (𝐶 ∈ Cat → (Inv‘𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
192, 18syl 17 . 2 (𝜑 → (Inv‘𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
201, 19syl5eq 2806 1 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cin 3714  ccnv 5265  cfv 6049  (class class class)co 6814  cmpt2 6816  Basecbs 16079  Catccat 16546  Sectcsect 16625  Invcinv 16626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-inv 16629
This theorem is referenced by:  invfval  16640  isoval  16646
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