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

Theorem funcinv 16356
 Description: The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
funcinv.b 𝐵 = (Base‘𝐷)
funcinv.s 𝐼 = (Inv‘𝐷)
funcinv.t 𝐽 = (Inv‘𝐸)
funcinv.f (𝜑𝐹(𝐷 Func 𝐸)𝐺)
funcinv.x (𝜑𝑋𝐵)
funcinv.y (𝜑𝑌𝐵)
funcinv.m (𝜑𝑀(𝑋𝐼𝑌)𝑁)
Assertion
Ref Expression
funcinv (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))

Proof of Theorem funcinv
StepHypRef Expression
1 funcinv.b . . 3 𝐵 = (Base‘𝐷)
2 eqid 2610 . . 3 (Sect‘𝐷) = (Sect‘𝐷)
3 eqid 2610 . . 3 (Sect‘𝐸) = (Sect‘𝐸)
4 funcinv.f . . 3 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
5 funcinv.x . . 3 (𝜑𝑋𝐵)
6 funcinv.y . . 3 (𝜑𝑌𝐵)
7 funcinv.m . . . . 5 (𝜑𝑀(𝑋𝐼𝑌)𝑁)
8 funcinv.s . . . . . 6 𝐼 = (Inv‘𝐷)
9 df-br 4584 . . . . . . . . 9 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
104, 9sylib 207 . . . . . . . 8 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11 funcrcl 16346 . . . . . . . 8 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1210, 11syl 17 . . . . . . 7 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1312simpld 474 . . . . . 6 (𝜑𝐷 ∈ Cat)
141, 8, 13, 5, 6, 2isinv 16243 . . . . 5 (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)))
157, 14mpbid 221 . . . 4 (𝜑 → (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁𝑁(𝑌(Sect‘𝐷)𝑋)𝑀))
1615simpld 474 . . 3 (𝜑𝑀(𝑋(Sect‘𝐷)𝑌)𝑁)
171, 2, 3, 4, 5, 6, 16funcsect 16355 . 2 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Sect‘𝐸)(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
1815simprd 478 . . 3 (𝜑𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)
191, 2, 3, 4, 6, 5, 18funcsect 16355 . 2 (𝜑 → ((𝑌𝐺𝑋)‘𝑁)((𝐹𝑌)(Sect‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀))
20 eqid 2610 . . 3 (Base‘𝐸) = (Base‘𝐸)
21 funcinv.t . . 3 𝐽 = (Inv‘𝐸)
2212simprd 478 . . 3 (𝜑𝐸 ∈ Cat)
231, 20, 4funcf1 16349 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐸))
2423, 5ffvelrnd 6268 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
2523, 6ffvelrnd 6268 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
2620, 21, 22, 24, 25, 3isinv 16243 . 2 (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Sect‘𝐸)(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁) ∧ ((𝑌𝐺𝑋)‘𝑁)((𝐹𝑌)(Sect‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀))))
2717, 19, 26mpbir2and 959 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Catccat 16148  Sectcsect 16227  Invcinv 16228   Func cfunc 16337 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ixp 7795  df-sect 16230  df-inv 16231  df-func 16341 This theorem is referenced by:  funciso  16357
 Copyright terms: Public domain W3C validator