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Theorem fthinv 17873
Description: A faithful functor reflects inverses. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fthsect.b 𝐵 = (Base‘𝐶)
fthsect.h 𝐻 = (Hom ‘𝐶)
fthsect.f (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
fthsect.x (𝜑𝑋𝐵)
fthsect.y (𝜑𝑌𝐵)
fthsect.m (𝜑𝑀 ∈ (𝑋𝐻𝑌))
fthsect.n (𝜑𝑁 ∈ (𝑌𝐻𝑋))
fthinv.s 𝐼 = (Inv‘𝐶)
fthinv.t 𝐽 = (Inv‘𝐷)
Assertion
Ref Expression
fthinv (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁)))

Proof of Theorem fthinv
StepHypRef Expression
1 fthsect.b . . . 4 𝐵 = (Base‘𝐶)
2 fthsect.h . . . 4 𝐻 = (Hom ‘𝐶)
3 fthsect.f . . . 4 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
4 fthsect.x . . . 4 (𝜑𝑋𝐵)
5 fthsect.y . . . 4 (𝜑𝑌𝐵)
6 fthsect.m . . . 4 (𝜑𝑀 ∈ (𝑋𝐻𝑌))
7 fthsect.n . . . 4 (𝜑𝑁 ∈ (𝑌𝐻𝑋))
8 eqid 2732 . . . 4 (Sect‘𝐶) = (Sect‘𝐶)
9 eqid 2732 . . . 4 (Sect‘𝐷) = (Sect‘𝐷)
101, 2, 3, 4, 5, 6, 7, 8, 9fthsect 17872 . . 3 (𝜑 → (𝑀(𝑋(Sect‘𝐶)𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Sect‘𝐷)(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁)))
111, 2, 3, 5, 4, 7, 6, 8, 9fthsect 17872 . . 3 (𝜑 → (𝑁(𝑌(Sect‘𝐶)𝑋)𝑀 ↔ ((𝑌𝐺𝑋)‘𝑁)((𝐹𝑌)(Sect‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)))
1210, 11anbi12d 631 . 2 (𝜑 → ((𝑀(𝑋(Sect‘𝐶)𝑌)𝑁𝑁(𝑌(Sect‘𝐶)𝑋)𝑀) ↔ (((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Sect‘𝐷)(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁) ∧ ((𝑌𝐺𝑋)‘𝑁)((𝐹𝑌)(Sect‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀))))
13 fthinv.s . . 3 𝐼 = (Inv‘𝐶)
14 fthfunc 17854 . . . . . . . 8 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
1514ssbri 5192 . . . . . . 7 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
163, 15syl 17 . . . . . 6 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
17 df-br 5148 . . . . . 6 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
1816, 17sylib 217 . . . . 5 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
19 funcrcl 17809 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2018, 19syl 17 . . . 4 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2120simpld 495 . . 3 (𝜑𝐶 ∈ Cat)
221, 13, 21, 4, 5, 8isinv 17703 . 2 (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐶)𝑌)𝑁𝑁(𝑌(Sect‘𝐶)𝑋)𝑀)))
23 eqid 2732 . . 3 (Base‘𝐷) = (Base‘𝐷)
24 fthinv.t . . 3 𝐽 = (Inv‘𝐷)
2520simprd 496 . . 3 (𝜑𝐷 ∈ Cat)
261, 23, 16funcf1 17812 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐷))
2726, 4ffvelcdmd 7084 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐷))
2826, 5ffvelcdmd 7084 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐷))
2923, 24, 25, 27, 28, 9isinv 17703 . 2 (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)(Sect‘𝐷)(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁) ∧ ((𝑌𝐺𝑋)‘𝑁)((𝐹𝑌)(Sect‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀))))
3012, 22, 293bitr4d 310 1 (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  cop 4633   class class class wbr 5147  cfv 6540  (class class class)co 7405  Basecbs 17140  Hom chom 17204  Catccat 17604  Sectcsect 17687  Invcinv 17688   Func cfunc 17800   Faith cfth 17850
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-ixp 8888  df-cat 17608  df-cid 17609  df-sect 17690  df-inv 17691  df-func 17804  df-fth 17852
This theorem is referenced by:  ffthiso  17876
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