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Theorem funcsect 16304
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 2610 . . . . . . 7 (Hom ‘𝐷) = (Hom ‘𝐷)
4 eqid 2610 . . . . . . 7 (comp‘𝐷) = (comp‘𝐷)
5 eqid 2610 . . . . . . 7 (Id‘𝐷) = (Id‘𝐷)
6 funcsect.s . . . . . . 7 𝑆 = (Sect‘𝐷)
7 funcsect.f . . . . . . . . . 10 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
8 df-br 4579 . . . . . . . . . 10 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
97, 8sylib 207 . . . . . . . . 9 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
10 funcrcl 16295 . . . . . . . . 9 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
119, 10syl 17 . . . . . . . 8 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1211simpld 474 . . . . . . 7 (𝜑𝐷 ∈ Cat)
13 funcsect.x . . . . . . 7 (𝜑𝑋𝐵)
14 funcsect.y . . . . . . 7 (𝜑𝑌𝐵)
152, 3, 4, 5, 6, 12, 13, 14issect 16185 . . . . . 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 221 . . . . 5 (𝜑 → (𝑀 ∈ (𝑋(Hom ‘𝐷)𝑌) ∧ 𝑁 ∈ (𝑌(Hom ‘𝐷)𝑋) ∧ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀) = ((Id‘𝐷)‘𝑋)))
1716simp3d 1068 . . . 4 (𝜑 → (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀) = ((Id‘𝐷)‘𝑋))
1817fveq2d 6092 . . 3 (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀)) = ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)))
19 eqid 2610 . . . 4 (comp‘𝐸) = (comp‘𝐸)
2016simp1d 1066 . . . 4 (𝜑𝑀 ∈ (𝑋(Hom ‘𝐷)𝑌))
2116simp2d 1067 . . . 4 (𝜑𝑁 ∈ (𝑌(Hom ‘𝐷)𝑋))
222, 3, 4, 19, 7, 13, 14, 13, 20, 21funcco 16303 . . 3 (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)))
23 eqid 2610 . . . 4 (Id‘𝐸) = (Id‘𝐸)
242, 5, 23, 7, 13funcid 16302 . . 3 (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)) = ((Id‘𝐸)‘(𝐹𝑋)))
2518, 22, 243eqtr3d 2652 . 2 (𝜑 → (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐸)‘(𝐹𝑋)))
26 eqid 2610 . . 3 (Base‘𝐸) = (Base‘𝐸)
27 eqid 2610 . . 3 (Hom ‘𝐸) = (Hom ‘𝐸)
28 funcsect.t . . 3 𝑇 = (Sect‘𝐸)
2911simprd 478 . . 3 (𝜑𝐸 ∈ Cat)
302, 26, 7funcf1 16298 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐸))
3130, 13ffvelrnd 6253 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
3230, 14ffvelrnd 6253 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
332, 3, 27, 7, 13, 14funcf2 16300 . . . 4 (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐷)𝑌)⟶((𝐹𝑋)(Hom ‘𝐸)(𝐹𝑌)))
3433, 20ffvelrnd 6253 . . 3 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)(Hom ‘𝐸)(𝐹𝑌)))
352, 3, 27, 7, 14, 13funcf2 16300 . . . 4 (𝜑 → (𝑌𝐺𝑋):(𝑌(Hom ‘𝐷)𝑋)⟶((𝐹𝑌)(Hom ‘𝐸)(𝐹𝑋)))
3635, 21ffvelrnd 6253 . . 3 (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹𝑌)(Hom ‘𝐸)(𝐹𝑋)))
3726, 27, 19, 23, 28, 29, 31, 32, 34, 36issect2 16186 . 2 (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐸)‘(𝐹𝑋))))
3825, 37mpbird 246 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  cop 4131   class class class wbr 4578  cfv 5790  (class class class)co 6527  Basecbs 15644  Hom chom 15728  compcco 15729  Catccat 16097  Idccid 16098  Sectcsect 16176   Func cfunc 16286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-map 7724  df-ixp 7773  df-sect 16179  df-func 16290
This theorem is referenced by:  funcinv  16305
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