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

Theorem funcinv 17143
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 2824 . . 3 (Sect‘𝐷) = (Sect‘𝐷)
3 eqid 2824 . . 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 5053 . . . . . . . . 9 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
104, 9sylib 221 . . . . . . . 8 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11 funcrcl 17133 . . . . . . . 8 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1210, 11syl 17 . . . . . . 7 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1312simpld 498 . . . . . 6 (𝜑𝐷 ∈ Cat)
141, 8, 13, 5, 6, 2isinv 17030 . . . . 5 (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)))
157, 14mpbid 235 . . . 4 (𝜑 → (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁𝑁(𝑌(Sect‘𝐷)𝑋)𝑀))
1615simpld 498 . . 3 (𝜑𝑀(𝑋(Sect‘𝐷)𝑌)𝑁)
171, 2, 3, 4, 5, 6, 16funcsect 17142 . 2 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Sect‘𝐸)(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
1815simprd 499 . . 3 (𝜑𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)
191, 2, 3, 4, 6, 5, 18funcsect 17142 . 2 (𝜑 → ((𝑌𝐺𝑋)‘𝑁)((𝐹𝑌)(Sect‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀))
20 eqid 2824 . . 3 (Base‘𝐸) = (Base‘𝐸)
21 funcinv.t . . 3 𝐽 = (Inv‘𝐸)
2212simprd 499 . . 3 (𝜑𝐸 ∈ Cat)
231, 20, 4funcf1 17136 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐸))
2423, 5ffvelrnd 6843 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
2523, 6ffvelrnd 6843 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
2620, 21, 22, 24, 25, 3isinv 17030 . 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 712 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cop 4556   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  Catccat 16935  Sectcsect 17014  Invcinv 17015   Func cfunc 17124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-map 8404  df-ixp 8458  df-sect 17017  df-inv 17018  df-func 17128
This theorem is referenced by:  funciso  17144
  Copyright terms: Public domain W3C validator