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Theorem funcsect 16738
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 2770 . . . . . . 7 (Hom ‘𝐷) = (Hom ‘𝐷)
4 eqid 2770 . . . . . . 7 (comp‘𝐷) = (comp‘𝐷)
5 eqid 2770 . . . . . . 7 (Id‘𝐷) = (Id‘𝐷)
6 funcsect.s . . . . . . 7 𝑆 = (Sect‘𝐷)
7 funcsect.f . . . . . . . . . 10 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
8 df-br 4785 . . . . . . . . . 10 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
97, 8sylib 208 . . . . . . . . 9 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
10 funcrcl 16729 . . . . . . . . 9 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
119, 10syl 17 . . . . . . . 8 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1211simpld 476 . . . . . . 7 (𝜑𝐷 ∈ Cat)
13 funcsect.x . . . . . . 7 (𝜑𝑋𝐵)
14 funcsect.y . . . . . . 7 (𝜑𝑌𝐵)
152, 3, 4, 5, 6, 12, 13, 14issect 16619 . . . . . 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 222 . . . . 5 (𝜑 → (𝑀 ∈ (𝑋(Hom ‘𝐷)𝑌) ∧ 𝑁 ∈ (𝑌(Hom ‘𝐷)𝑋) ∧ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀) = ((Id‘𝐷)‘𝑋)))
1716simp3d 1137 . . . 4 (𝜑 → (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀) = ((Id‘𝐷)‘𝑋))
1817fveq2d 6336 . . 3 (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀)) = ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)))
19 eqid 2770 . . . 4 (comp‘𝐸) = (comp‘𝐸)
2016simp1d 1135 . . . 4 (𝜑𝑀 ∈ (𝑋(Hom ‘𝐷)𝑌))
2116simp2d 1136 . . . 4 (𝜑𝑁 ∈ (𝑌(Hom ‘𝐷)𝑋))
222, 3, 4, 19, 7, 13, 14, 13, 20, 21funcco 16737 . . 3 (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)))
23 eqid 2770 . . . 4 (Id‘𝐸) = (Id‘𝐸)
242, 5, 23, 7, 13funcid 16736 . . 3 (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)) = ((Id‘𝐸)‘(𝐹𝑋)))
2518, 22, 243eqtr3d 2812 . 2 (𝜑 → (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐸)‘(𝐹𝑋)))
26 eqid 2770 . . 3 (Base‘𝐸) = (Base‘𝐸)
27 eqid 2770 . . 3 (Hom ‘𝐸) = (Hom ‘𝐸)
28 funcsect.t . . 3 𝑇 = (Sect‘𝐸)
2911simprd 477 . . 3 (𝜑𝐸 ∈ Cat)
302, 26, 7funcf1 16732 . . . 4 (𝜑𝐹:𝐵⟶(Base‘𝐸))
3130, 13ffvelrnd 6503 . . 3 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐸))
3230, 14ffvelrnd 6503 . . 3 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐸))
332, 3, 27, 7, 13, 14funcf2 16734 . . . 4 (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐷)𝑌)⟶((𝐹𝑋)(Hom ‘𝐸)(𝐹𝑌)))
3433, 20ffvelrnd 6503 . . 3 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)(Hom ‘𝐸)(𝐹𝑌)))
352, 3, 27, 7, 14, 13funcf2 16734 . . . 4 (𝜑 → (𝑌𝐺𝑋):(𝑌(Hom ‘𝐷)𝑋)⟶((𝐹𝑌)(Hom ‘𝐸)(𝐹𝑋)))
3635, 21ffvelrnd 6503 . . 3 (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹𝑌)(Hom ‘𝐸)(𝐹𝑋)))
3726, 27, 19, 23, 28, 29, 31, 32, 34, 36issect2 16620 . 2 (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐸)‘(𝐹𝑋))))
3825, 37mpbird 247 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  cop 4320   class class class wbr 4784  cfv 6031  (class class class)co 6792  Basecbs 16063  Hom chom 16159  compcco 16160  Catccat 16531  Idccid 16532  Sectcsect 16610   Func cfunc 16720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-map 8010  df-ixp 8062  df-sect 16613  df-func 16724
This theorem is referenced by:  funcinv  16739
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