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Theorem funcinv 17719
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 2737 . . 3 (Sect‘𝐷) = (Sect‘𝐷)
3 eqid 2737 . . 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 5104 . . . . . . . . 9 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
104, 9sylib 217 . . . . . . . 8 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11 funcrcl 17709 . . . . . . . 8 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1210, 11syl 17 . . . . . . 7 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1312simpld 495 . . . . . 6 (𝜑𝐷 ∈ Cat)
141, 8, 13, 5, 6, 2isinv 17603 . . . . 5 (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)))
157, 14mpbid 231 . . . 4 (𝜑 → (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁𝑁(𝑌(Sect‘𝐷)𝑋)𝑀))
1615simpld 495 . . 3 (𝜑𝑀(𝑋(Sect‘𝐷)𝑌)𝑁)
171, 2, 3, 4, 5, 6, 16funcsect 17718 . 2 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Sect‘𝐸)(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
1815simprd 496 . . 3 (𝜑𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)
191, 2, 3, 4, 6, 5, 18funcsect 17718 . 2 (𝜑 → ((𝑌𝐺𝑋)‘𝑁)((𝐹𝑌)(Sect‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀))
20 eqid 2737 . . 3 (Base‘𝐸) = (Base‘𝐸)
21 funcinv.t . . 3 𝐽 = (Inv‘𝐸)
2212simprd 496 . . 3 (𝜑𝐸 ∈ Cat)
231, 20, 4funcf1 17712 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐸))
2423, 5ffvelcdmd 7032 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
2523, 6ffvelcdmd 7032 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
2620, 21, 22, 24, 25, 3isinv 17603 . 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 711 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cop 4590   class class class wbr 5103  cfv 6493  (class class class)co 7351  Basecbs 17043  Catccat 17504  Sectcsect 17587  Invcinv 17588   Func cfunc 17700
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-map 8725  df-ixp 8794  df-sect 17590  df-inv 17591  df-func 17704
This theorem is referenced by:  funciso  17720
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