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Theorem funcinv 17937
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 2740 . . 3 (Sect‘𝐷) = (Sect‘𝐷)
3 eqid 2740 . . 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 5167 . . . . . . . . 9 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
104, 9sylib 218 . . . . . . . 8 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11 funcrcl 17927 . . . . . . . 8 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1210, 11syl 17 . . . . . . 7 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1312simpld 494 . . . . . 6 (𝜑𝐷 ∈ Cat)
141, 8, 13, 5, 6, 2isinv 17821 . . . . 5 (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)))
157, 14mpbid 232 . . . 4 (𝜑 → (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁𝑁(𝑌(Sect‘𝐷)𝑋)𝑀))
1615simpld 494 . . 3 (𝜑𝑀(𝑋(Sect‘𝐷)𝑌)𝑁)
171, 2, 3, 4, 5, 6, 16funcsect 17936 . 2 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Sect‘𝐸)(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
1815simprd 495 . . 3 (𝜑𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)
191, 2, 3, 4, 6, 5, 18funcsect 17936 . 2 (𝜑 → ((𝑌𝐺𝑋)‘𝑁)((𝐹𝑌)(Sect‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀))
20 eqid 2740 . . 3 (Base‘𝐸) = (Base‘𝐸)
21 funcinv.t . . 3 𝐽 = (Inv‘𝐸)
2212simprd 495 . . 3 (𝜑𝐸 ∈ Cat)
231, 20, 4funcf1 17930 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐸))
2423, 5ffvelcdmd 7119 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
2523, 6ffvelcdmd 7119 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
2620, 21, 22, 24, 25, 3isinv 17821 . 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 395   = wceq 1537  wcel 2108  cop 4654   class class class wbr 5166  cfv 6573  (class class class)co 7448  Basecbs 17258  Catccat 17722  Sectcsect 17805  Invcinv 17806   Func cfunc 17918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-map 8886  df-ixp 8956  df-sect 17808  df-inv 17809  df-func 17922
This theorem is referenced by:  funciso  17938
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