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Theorem fthsect 16937
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 2825 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
4 fthsect.f . . . 4 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
5 fthsect.x . . . 4 (𝜑𝑋𝐵)
6 eqid 2825 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
7 fthfunc 16919 . . . . . . . . . 10 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
87ssbri 4918 . . . . . . . . 9 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
94, 8syl 17 . . . . . . . 8 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
10 df-br 4874 . . . . . . . 8 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylib 210 . . . . . . 7 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12 funcrcl 16875 . . . . . . 7 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1311, 12syl 17 . . . . . 6 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1413simpld 490 . . . . 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 16698 . . . 4 (𝜑 → (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) ∈ (𝑋𝐻𝑋))
19 eqid 2825 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
201, 2, 19, 14, 5catidcl 16695 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
211, 2, 3, 4, 5, 5, 18, 20fthi 16930 . . 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 2825 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
231, 2, 6, 22, 9, 5, 15, 5, 16, 17funcco 16883 . . . 4 (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)))
24 eqid 2825 . . . . 5 (Id‘𝐷) = (Id‘𝐷)
251, 19, 24, 9, 5funcid 16882 . . . 4 (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐶)‘𝑋)) = ((Id‘𝐷)‘(𝐹𝑋)))
2623, 25eqeq12d 2840 . . 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 273 . 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 16766 . 2 (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) = ((Id‘𝐶)‘𝑋)))
30 eqid 2825 . . 3 (Base‘𝐷) = (Base‘𝐷)
31 fthsect.t . . 3 𝑇 = (Sect‘𝐷)
3213simprd 491 . . 3 (𝜑𝐷 ∈ Cat)
331, 30, 9funcf1 16878 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐷))
3433, 5ffvelrnd 6609 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐷))
3533, 15ffvelrnd 6609 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐷))
361, 2, 3, 9, 5, 15funcf2 16880 . . . 4 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)(Hom ‘𝐷)(𝐹𝑌)))
3736, 16ffvelrnd 6609 . . 3 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)(Hom ‘𝐷)(𝐹𝑌)))
381, 2, 3, 9, 15, 5funcf2 16880 . . . 4 (𝜑 → (𝑌𝐺𝑋):(𝑌𝐻𝑋)⟶((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
3938, 17ffvelrnd 6609 . . 3 (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
4030, 3, 22, 24, 31, 32, 34, 35, 37, 39issect2 16766 . 2 (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐷)‘(𝐹𝑋))))
4127, 29, 403bitr4d 303 1 (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  cop 4403   class class class wbr 4873  cfv 6123  (class class class)co 6905  Basecbs 16222  Hom chom 16316  compcco 16317  Catccat 16677  Idccid 16678  Sectcsect 16756   Func cfunc 16866   Faith cfth 16915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-map 8124  df-ixp 8176  df-cat 16681  df-cid 16682  df-sect 16759  df-func 16870  df-fth 16917
This theorem is referenced by:  fthinv  16938
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