MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  invfval Structured version   Visualization version   GIF version

Theorem invfval 16620
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 16619 . 2 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
7 simprl 811 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
8 simprr 813 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
97, 8oveq12d 6831 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝑆𝑦) = (𝑋𝑆𝑌))
108, 7oveq12d 6831 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋))
1110cnveqd 5453 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋))
129, 11ineq12d 3958 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥)) = ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)))
13 invfval.y . 2 (𝜑𝑌𝐵)
14 ovex 6841 . . . 4 (𝑋𝑆𝑌) ∈ V
1514inex1 4951 . . 3 ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)) ∈ V
1615a1i 11 . 2 (𝜑 → ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)) ∈ V)
176, 12, 4, 13, 16ovmpt2d 6953 1 (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cin 3714  ccnv 5265  cfv 6049  (class class class)co 6813  Basecbs 16059  Catccat 16526  Sectcsect 16605  Invcinv 16606
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 7114
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 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-inv 16609
This theorem is referenced by:  isinv  16621  invss  16622  dfiso2  16633  oppcinv  16641
  Copyright terms: Public domain W3C validator