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Theorem fthsect 16408
Description: A faithful functor reflects sections. (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))
fthsect.s 𝑆 = (Sect‘𝐶)
fthsect.t 𝑇 = (Sect‘𝐷)
Assertion
Ref Expression
fthsect (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁)))

Proof of Theorem fthsect
StepHypRef Expression
1 fthsect.b . . . 4 𝐵 = (Base‘𝐶)
2 fthsect.h . . . 4 𝐻 = (Hom ‘𝐶)
3 eqid 2610 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
4 fthsect.f . . . 4 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
5 fthsect.x . . . 4 (𝜑𝑋𝐵)
6 eqid 2610 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
7 fthfunc 16390 . . . . . . . . . 10 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
87ssbri 4627 . . . . . . . . 9 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
94, 8syl 17 . . . . . . . 8 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
10 df-br 4584 . . . . . . . 8 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylib 207 . . . . . . 7 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12 funcrcl 16346 . . . . . . 7 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1311, 12syl 17 . . . . . 6 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1413simpld 474 . . . . 5 (𝜑𝐶 ∈ Cat)
15 fthsect.y . . . . 5 (𝜑𝑌𝐵)
16 fthsect.m . . . . 5 (𝜑𝑀 ∈ (𝑋𝐻𝑌))
17 fthsect.n . . . . 5 (𝜑𝑁 ∈ (𝑌𝐻𝑋))
181, 2, 6, 14, 5, 15, 5, 16, 17catcocl 16169 . . . 4 (𝜑 → (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) ∈ (𝑋𝐻𝑋))
19 eqid 2610 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
201, 2, 19, 14, 5catidcl 16166 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
211, 2, 3, 4, 5, 5, 18, 20fthi 16401 . . 3 (𝜑 → (((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀)) = ((𝑋𝐺𝑋)‘((Id‘𝐶)‘𝑋)) ↔ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) = ((Id‘𝐶)‘𝑋)))
22 eqid 2610 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
231, 2, 6, 22, 9, 5, 15, 5, 16, 17funcco 16354 . . . 4 (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)))
24 eqid 2610 . . . . 5 (Id‘𝐷) = (Id‘𝐷)
251, 19, 24, 9, 5funcid 16353 . . . 4 (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐶)‘𝑋)) = ((Id‘𝐷)‘(𝐹𝑋)))
2623, 25eqeq12d 2625 . . 3 (𝜑 → (((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀)) = ((𝑋𝐺𝑋)‘((Id‘𝐶)‘𝑋)) ↔ (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐷)‘(𝐹𝑋))))
2721, 26bitr3d 269 . 2 (𝜑 → ((𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) = ((Id‘𝐶)‘𝑋) ↔ (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐷)‘(𝐹𝑋))))
28 fthsect.s . . 3 𝑆 = (Sect‘𝐶)
291, 2, 6, 19, 28, 14, 5, 15, 16, 17issect2 16237 . 2 (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) = ((Id‘𝐶)‘𝑋)))
30 eqid 2610 . . 3 (Base‘𝐷) = (Base‘𝐷)
31 fthsect.t . . 3 𝑇 = (Sect‘𝐷)
3213simprd 478 . . 3 (𝜑𝐷 ∈ Cat)
331, 30, 9funcf1 16349 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐷))
3433, 5ffvelrnd 6268 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐷))
3533, 15ffvelrnd 6268 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐷))
361, 2, 3, 9, 5, 15funcf2 16351 . . . 4 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)(Hom ‘𝐷)(𝐹𝑌)))
3736, 16ffvelrnd 6268 . . 3 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)(Hom ‘𝐷)(𝐹𝑌)))
381, 2, 3, 9, 15, 5funcf2 16351 . . . 4 (𝜑 → (𝑌𝐺𝑋):(𝑌𝐻𝑋)⟶((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
3938, 17ffvelrnd 6268 . . 3 (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
4030, 3, 22, 24, 31, 32, 34, 35, 37, 39issect2 16237 . 2 (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐷)‘(𝐹𝑋))))
4127, 29, 403bitr4d 299 1 (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  cop 4131   class class class wbr 4583  cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149  Sectcsect 16227   Func cfunc 16337   Faith cfth 16386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ixp 7795  df-cat 16152  df-cid 16153  df-sect 16230  df-func 16341  df-fth 16388
This theorem is referenced by:  fthinv  16409
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