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Theorem funcsect 17938
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 2740 . . . . . . 7 (Hom ‘𝐷) = (Hom ‘𝐷)
4 eqid 2740 . . . . . . 7 (comp‘𝐷) = (comp‘𝐷)
5 eqid 2740 . . . . . . 7 (Id‘𝐷) = (Id‘𝐷)
6 funcsect.s . . . . . . 7 𝑆 = (Sect‘𝐷)
7 funcsect.f . . . . . . . . . 10 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
8 df-br 5167 . . . . . . . . . 10 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
97, 8sylib 218 . . . . . . . . 9 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
10 funcrcl 17929 . . . . . . . . 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 17816 . . . . . 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 6926 . . 3 (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀)) = ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)))
19 eqid 2740 . . . 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 17937 . . 3 (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)))
23 eqid 2740 . . . 4 (Id‘𝐸) = (Id‘𝐸)
242, 5, 23, 7, 13funcid 17936 . . 3 (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)) = ((Id‘𝐸)‘(𝐹𝑋)))
2518, 22, 243eqtr3d 2788 . 2 (𝜑 → (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐸)‘(𝐹𝑋)))
26 eqid 2740 . . 3 (Base‘𝐸) = (Base‘𝐸)
27 eqid 2740 . . 3 (Hom ‘𝐸) = (Hom ‘𝐸)
28 funcsect.t . . 3 𝑇 = (Sect‘𝐸)
2911simprd 495 . . 3 (𝜑𝐸 ∈ Cat)
302, 26, 7funcf1 17932 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐸))
3130, 13ffvelcdmd 7121 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
3230, 14ffvelcdmd 7121 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
332, 3, 27, 7, 13, 14funcf2 17934 . . . 4 (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐷)𝑌)⟶((𝐹𝑋)(Hom ‘𝐸)(𝐹𝑌)))
3433, 20ffvelcdmd 7121 . . 3 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)(Hom ‘𝐸)(𝐹𝑌)))
352, 3, 27, 7, 14, 13funcf2 17934 . . . 4 (𝜑 → (𝑌𝐺𝑋):(𝑌(Hom ‘𝐷)𝑋)⟶((𝐹𝑌)(Hom ‘𝐸)(𝐹𝑋)))
3635, 21ffvelcdmd 7121 . . 3 (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹𝑌)(Hom ‘𝐸)(𝐹𝑋)))
3726, 27, 19, 23, 28, 29, 31, 32, 34, 36issect2 17817 . 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 2108  cop 4654   class class class wbr 5166  cfv 6575  (class class class)co 7450  Basecbs 17260  Hom chom 17324  compcco 17325  Catccat 17724  Idccid 17725  Sectcsect 17807   Func cfunc 17920
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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772
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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-1st 8032  df-2nd 8033  df-map 8888  df-ixp 8958  df-sect 17810  df-func 17924
This theorem is referenced by:  funcinv  17939
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