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Theorem funcsect 17931
Description: The image of a section under a functor is a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
funcsect.b 𝐵 = (Base‘𝐷)
funcsect.s 𝑆 = (Sect‘𝐷)
funcsect.t 𝑇 = (Sect‘𝐸)
funcsect.f (𝜑𝐹(𝐷 Func 𝐸)𝐺)
funcsect.x (𝜑𝑋𝐵)
funcsect.y (𝜑𝑌𝐵)
funcsect.m (𝜑𝑀(𝑋𝑆𝑌)𝑁)
Assertion
Ref Expression
funcsect (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))

Proof of Theorem funcsect
StepHypRef Expression
1 funcsect.m . . . . . 6 (𝜑𝑀(𝑋𝑆𝑌)𝑁)
2 funcsect.b . . . . . . 7 𝐵 = (Base‘𝐷)
3 eqid 2734 . . . . . . 7 (Hom ‘𝐷) = (Hom ‘𝐷)
4 eqid 2734 . . . . . . 7 (comp‘𝐷) = (comp‘𝐷)
5 eqid 2734 . . . . . . 7 (Id‘𝐷) = (Id‘𝐷)
6 funcsect.s . . . . . . 7 𝑆 = (Sect‘𝐷)
7 funcsect.f . . . . . . . . . 10 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
8 df-br 5170 . . . . . . . . . 10 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
97, 8sylib 218 . . . . . . . . 9 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
10 funcrcl 17922 . . . . . . . . 9 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
119, 10syl 17 . . . . . . . 8 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1211simpld 494 . . . . . . 7 (𝜑𝐷 ∈ Cat)
13 funcsect.x . . . . . . 7 (𝜑𝑋𝐵)
14 funcsect.y . . . . . . 7 (𝜑𝑌𝐵)
152, 3, 4, 5, 6, 12, 13, 14issect 17809 . . . . . 6 (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ (𝑀 ∈ (𝑋(Hom ‘𝐷)𝑌) ∧ 𝑁 ∈ (𝑌(Hom ‘𝐷)𝑋) ∧ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀) = ((Id‘𝐷)‘𝑋))))
161, 15mpbid 232 . . . . 5 (𝜑 → (𝑀 ∈ (𝑋(Hom ‘𝐷)𝑌) ∧ 𝑁 ∈ (𝑌(Hom ‘𝐷)𝑋) ∧ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀) = ((Id‘𝐷)‘𝑋)))
1716simp3d 1144 . . . 4 (𝜑 → (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀) = ((Id‘𝐷)‘𝑋))
1817fveq2d 6923 . . 3 (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀)) = ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)))
19 eqid 2734 . . . 4 (comp‘𝐸) = (comp‘𝐸)
2016simp1d 1142 . . . 4 (𝜑𝑀 ∈ (𝑋(Hom ‘𝐷)𝑌))
2116simp2d 1143 . . . 4 (𝜑𝑁 ∈ (𝑌(Hom ‘𝐷)𝑋))
222, 3, 4, 19, 7, 13, 14, 13, 20, 21funcco 17930 . . 3 (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)))
23 eqid 2734 . . . 4 (Id‘𝐸) = (Id‘𝐸)
242, 5, 23, 7, 13funcid 17929 . . 3 (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)) = ((Id‘𝐸)‘(𝐹𝑋)))
2518, 22, 243eqtr3d 2782 . 2 (𝜑 → (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐸)‘(𝐹𝑋)))
26 eqid 2734 . . 3 (Base‘𝐸) = (Base‘𝐸)
27 eqid 2734 . . 3 (Hom ‘𝐸) = (Hom ‘𝐸)
28 funcsect.t . . 3 𝑇 = (Sect‘𝐸)
2911simprd 495 . . 3 (𝜑𝐸 ∈ Cat)
302, 26, 7funcf1 17925 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐸))
3130, 13ffvelcdmd 7117 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
3230, 14ffvelcdmd 7117 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
332, 3, 27, 7, 13, 14funcf2 17927 . . . 4 (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐷)𝑌)⟶((𝐹𝑋)(Hom ‘𝐸)(𝐹𝑌)))
3433, 20ffvelcdmd 7117 . . 3 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)(Hom ‘𝐸)(𝐹𝑌)))
352, 3, 27, 7, 14, 13funcf2 17927 . . . 4 (𝜑 → (𝑌𝐺𝑋):(𝑌(Hom ‘𝐷)𝑋)⟶((𝐹𝑌)(Hom ‘𝐸)(𝐹𝑋)))
3635, 21ffvelcdmd 7117 . . 3 (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹𝑌)(Hom ‘𝐸)(𝐹𝑋)))
3726, 27, 19, 23, 28, 29, 31, 32, 34, 36issect2 17810 . 2 (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐸)‘(𝐹𝑋))))
3825, 37mpbird 257 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1537  wcel 2103  cop 4654   class class class wbr 5169  cfv 6572  (class class class)co 7445  Basecbs 17253  Hom chom 17317  compcco 17318  Catccat 17717  Idccid 17718  Sectcsect 17800   Func cfunc 17913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-1st 8026  df-2nd 8027  df-map 8882  df-ixp 8952  df-sect 17803  df-func 17917
This theorem is referenced by:  funcinv  17932
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