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Theorem fthsect 17187
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 2819 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
4 fthsect.f . . . 4 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
5 fthsect.x . . . 4 (𝜑𝑋𝐵)
6 eqid 2819 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
7 fthfunc 17169 . . . . . . . . . 10 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
87ssbri 5102 . . . . . . . . 9 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
94, 8syl 17 . . . . . . . 8 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
10 df-br 5058 . . . . . . . 8 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylib 220 . . . . . . 7 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12 funcrcl 17125 . . . . . . 7 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1311, 12syl 17 . . . . . 6 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1413simpld 497 . . . . 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 16948 . . . 4 (𝜑 → (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) ∈ (𝑋𝐻𝑋))
19 eqid 2819 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
201, 2, 19, 14, 5catidcl 16945 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
211, 2, 3, 4, 5, 5, 18, 20fthi 17180 . . 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 2819 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
231, 2, 6, 22, 9, 5, 15, 5, 16, 17funcco 17133 . . . 4 (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)))
24 eqid 2819 . . . . 5 (Id‘𝐷) = (Id‘𝐷)
251, 19, 24, 9, 5funcid 17132 . . . 4 (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐶)‘𝑋)) = ((Id‘𝐷)‘(𝐹𝑋)))
2623, 25eqeq12d 2835 . . 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 283 . 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 17016 . 2 (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) = ((Id‘𝐶)‘𝑋)))
30 eqid 2819 . . 3 (Base‘𝐷) = (Base‘𝐷)
31 fthsect.t . . 3 𝑇 = (Sect‘𝐷)
3213simprd 498 . . 3 (𝜑𝐷 ∈ Cat)
331, 30, 9funcf1 17128 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐷))
3433, 5ffvelrnd 6845 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐷))
3533, 15ffvelrnd 6845 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐷))
361, 2, 3, 9, 5, 15funcf2 17130 . . . 4 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)(Hom ‘𝐷)(𝐹𝑌)))
3736, 16ffvelrnd 6845 . . 3 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)(Hom ‘𝐷)(𝐹𝑌)))
381, 2, 3, 9, 15, 5funcf2 17130 . . . 4 (𝜑 → (𝑌𝐺𝑋):(𝑌𝐻𝑋)⟶((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
3938, 17ffvelrnd 6845 . . 3 (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
4030, 3, 22, 24, 31, 32, 34, 35, 37, 39issect2 17016 . 2 (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐷)‘(𝐹𝑋))))
4127, 29, 403bitr4d 313 1 (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  cop 4565   class class class wbr 5057  cfv 6348  (class class class)co 7148  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927  Idccid 16928  Sectcsect 17006   Func cfunc 17116   Faith cfth 17165
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-map 8400  df-ixp 8454  df-cat 16931  df-cid 16932  df-sect 17009  df-func 17120  df-fth 17167
This theorem is referenced by:  fthinv  17188
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