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Theorem fthsect 17817
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 2733 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
4 fthsect.f . . . 4 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
5 fthsect.x . . . 4 (𝜑𝑋𝐵)
6 eqid 2733 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
7 fthfunc 17799 . . . . . . . . . 10 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
87ssbri 5151 . . . . . . . . 9 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
94, 8syl 17 . . . . . . . 8 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
10 df-br 5107 . . . . . . . 8 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylib 217 . . . . . . 7 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12 funcrcl 17754 . . . . . . 7 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1311, 12syl 17 . . . . . 6 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1413simpld 496 . . . . 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 17570 . . . 4 (𝜑 → (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) ∈ (𝑋𝐻𝑋))
19 eqid 2733 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
201, 2, 19, 14, 5catidcl 17567 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
211, 2, 3, 4, 5, 5, 18, 20fthi 17810 . . 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 2733 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
231, 2, 6, 22, 9, 5, 15, 5, 16, 17funcco 17762 . . . 4 (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐷)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)))
24 eqid 2733 . . . . 5 (Id‘𝐷) = (Id‘𝐷)
251, 19, 24, 9, 5funcid 17761 . . . 4 (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐶)‘𝑋)) = ((Id‘𝐷)‘(𝐹𝑋)))
2623, 25eqeq12d 2749 . . 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 17642 . 2 (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) = ((Id‘𝐶)‘𝑋)))
30 eqid 2733 . . 3 (Base‘𝐷) = (Base‘𝐷)
31 fthsect.t . . 3 𝑇 = (Sect‘𝐷)
3213simprd 497 . . 3 (𝜑𝐷 ∈ Cat)
331, 30, 9funcf1 17757 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐷))
3433, 5ffvelcdmd 7037 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐷))
3533, 15ffvelcdmd 7037 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐷))
361, 2, 3, 9, 5, 15funcf2 17759 . . . 4 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)(Hom ‘𝐷)(𝐹𝑌)))
3736, 16ffvelcdmd 7037 . . 3 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)(Hom ‘𝐷)(𝐹𝑌)))
381, 2, 3, 9, 15, 5funcf2 17759 . . . 4 (𝜑 → (𝑌𝐺𝑋):(𝑌𝐻𝑋)⟶((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
3938, 17ffvelcdmd 7037 . . 3 (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
4030, 3, 22, 24, 31, 32, 34, 35, 37, 39issect2 17642 . 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 397   = wceq 1542  wcel 2107  cop 4593   class class class wbr 5106  cfv 6497  (class class class)co 7358  Basecbs 17088  Hom chom 17149  compcco 17150  Catccat 17549  Idccid 17550  Sectcsect 17632   Func cfunc 17745   Faith cfth 17795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770  df-ixp 8839  df-cat 17553  df-cid 17554  df-sect 17635  df-func 17749  df-fth 17797
This theorem is referenced by:  fthinv  17818
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