MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fthsect Structured version   Visualization version   GIF version

Theorem fthsect 17884
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 2726 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
4 fthsect.f . . . 4 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
5 fthsect.x . . . 4 (𝜑𝑋𝐵)
6 eqid 2726 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
7 fthfunc 17866 . . . . . . . . . 10 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
87ssbri 5186 . . . . . . . . 9 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
94, 8syl 17 . . . . . . . 8 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
10 df-br 5142 . . . . . . . 8 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylib 217 . . . . . . 7 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12 funcrcl 17819 . . . . . . 7 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1311, 12syl 17 . . . . . 6 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1413simpld 494 . . . . 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 17635 . . . 4 (𝜑 → (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) ∈ (𝑋𝐻𝑋))
19 eqid 2726 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
201, 2, 19, 14, 5catidcl 17632 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
211, 2, 3, 4, 5, 5, 18, 20fthi 17877 . . 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 2726 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
231, 2, 6, 22, 9, 5, 15, 5, 16, 17funcco 17827 . . . 4 (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)))
24 eqid 2726 . . . . 5 (Id‘𝐷) = (Id‘𝐷)
251, 19, 24, 9, 5funcid 17826 . . . 4 (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐶)‘𝑋)) = ((Id‘𝐷)‘(𝐹𝑋)))
2623, 25eqeq12d 2742 . . 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 281 . 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 17707 . 2 (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) = ((Id‘𝐶)‘𝑋)))
30 eqid 2726 . . 3 (Base‘𝐷) = (Base‘𝐷)
31 fthsect.t . . 3 𝑇 = (Sect‘𝐷)
3213simprd 495 . . 3 (𝜑𝐷 ∈ Cat)
331, 30, 9funcf1 17822 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐷))
3433, 5ffvelcdmd 7080 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐷))
3533, 15ffvelcdmd 7080 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐷))
361, 2, 3, 9, 5, 15funcf2 17824 . . . 4 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)(Hom ‘𝐷)(𝐹𝑌)))
3736, 16ffvelcdmd 7080 . . 3 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)(Hom ‘𝐷)(𝐹𝑌)))
381, 2, 3, 9, 15, 5funcf2 17824 . . . 4 (𝜑 → (𝑌𝐺𝑋):(𝑌𝐻𝑋)⟶((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
3938, 17ffvelcdmd 7080 . . 3 (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
4030, 3, 22, 24, 31, 32, 34, 35, 37, 39issect2 17707 . 2 (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐷)‘(𝐹𝑋))))
4127, 29, 403bitr4d 311 1 (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  cop 4629   class class class wbr 5141  cfv 6536  (class class class)co 7404  Basecbs 17150  Hom chom 17214  compcco 17215  Catccat 17614  Idccid 17615  Sectcsect 17697   Func cfunc 17810   Faith cfth 17862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-ixp 8891  df-cat 17618  df-cid 17619  df-sect 17700  df-func 17814  df-fth 17864
This theorem is referenced by:  fthinv  17885
  Copyright terms: Public domain W3C validator