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Theorem funcinv 16739
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 2771 . . 3 (Sect‘𝐷) = (Sect‘𝐷)
3 eqid 2771 . . 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 4787 . . . . . . . . 9 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
104, 9sylib 208 . . . . . . . 8 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11 funcrcl 16729 . . . . . . . 8 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1210, 11syl 17 . . . . . . 7 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1312simpld 476 . . . . . 6 (𝜑𝐷 ∈ Cat)
141, 8, 13, 5, 6, 2isinv 16626 . . . . 5 (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)))
157, 14mpbid 222 . . . 4 (𝜑 → (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁𝑁(𝑌(Sect‘𝐷)𝑋)𝑀))
1615simpld 476 . . 3 (𝜑𝑀(𝑋(Sect‘𝐷)𝑌)𝑁)
171, 2, 3, 4, 5, 6, 16funcsect 16738 . 2 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Sect‘𝐸)(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
1815simprd 477 . . 3 (𝜑𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)
191, 2, 3, 4, 6, 5, 18funcsect 16738 . 2 (𝜑 → ((𝑌𝐺𝑋)‘𝑁)((𝐹𝑌)(Sect‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀))
20 eqid 2771 . . 3 (Base‘𝐸) = (Base‘𝐸)
21 funcinv.t . . 3 𝐽 = (Inv‘𝐸)
2212simprd 477 . . 3 (𝜑𝐸 ∈ Cat)
231, 20, 4funcf1 16732 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐸))
2423, 5ffvelrnd 6503 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
2523, 6ffvelrnd 6503 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
2620, 21, 22, 24, 25, 3isinv 16626 . 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 684 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cop 4322   class class class wbr 4786  cfv 6031  (class class class)co 6792  Basecbs 16063  Catccat 16531  Sectcsect 16610  Invcinv 16611   Func cfunc 16720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-map 8010  df-ixp 8062  df-sect 16613  df-inv 16614  df-func 16724
This theorem is referenced by:  funciso  16740
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