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Theorem fthinv 16355
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 2609 . . . 4 (Sect‘𝐶) = (Sect‘𝐶)
9 eqid 2609 . . . 4 (Sect‘𝐷) = (Sect‘𝐷)
101, 2, 3, 4, 5, 6, 7, 8, 9fthsect 16354 . . 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 16354 . . 3 (𝜑 → (𝑁(𝑌(Sect‘𝐶)𝑋)𝑀 ↔ ((𝑌𝐺𝑋)‘𝑁)((𝐹𝑌)(Sect‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)))
1210, 11anbi12d 742 . 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 16336 . . . . . . . 8 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
1514ssbri 4621 . . . . . . 7 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
163, 15syl 17 . . . . . 6 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
17 df-br 4578 . . . . . 6 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
1816, 17sylib 206 . . . . 5 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
19 funcrcl 16292 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2018, 19syl 17 . . . 4 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2120simpld 473 . . 3 (𝜑𝐶 ∈ Cat)
221, 13, 21, 4, 5, 8isinv 16189 . 2 (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐶)𝑌)𝑁𝑁(𝑌(Sect‘𝐶)𝑋)𝑀)))
23 eqid 2609 . . 3 (Base‘𝐷) = (Base‘𝐷)
24 fthinv.t . . 3 𝐽 = (Inv‘𝐷)
2520simprd 477 . . 3 (𝜑𝐷 ∈ Cat)
261, 23, 16funcf1 16295 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐷))
2726, 4ffvelrnd 6253 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐷))
2826, 5ffvelrnd 6253 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐷))
2923, 24, 25, 27, 28, 9isinv 16189 . 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 298 1 (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  cop 4130   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  Hom chom 15725  Catccat 16094  Sectcsect 16173  Invcinv 16174   Func cfunc 16283   Faith cfth 16332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-map 7723  df-ixp 7772  df-cat 16098  df-cid 16099  df-sect 16176  df-inv 16177  df-func 16287  df-fth 16334
This theorem is referenced by:  ffthiso  16358
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