Theorem funcsect 17826

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	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
funcsect.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
funcsect.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
funcsect.m	⊢ (𝜑 → 𝑀(𝑋𝑆𝑌)𝑁)

Assertion

Ref	Expression
funcsect	⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝑇(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))

Proof of Theorem funcsect

Step	Hyp	Ref	Expression
1		funcsect.m	. . . . . 6 ⊢ (𝜑 → 𝑀(𝑋𝑆𝑌)𝑁)
2		funcsect.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐷)
3		eqid 2732	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2732	. . . . . . 7 ⊢ (comp‘𝐷) = (comp‘𝐷)
5		eqid 2732	. . . . . . 7 ⊢ (Id‘𝐷) = (Id‘𝐷)
6		funcsect.s	. . . . . . 7 ⊢ 𝑆 = (Sect‘𝐷)
7		funcsect.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
8		df-br 5149	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
9	7, 8	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
10		funcrcl 17817	. . . . . . . . 9 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
11	9, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
12	11	simpld 495	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
13		funcsect.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
14		funcsect.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
15	2, 3, 4, 5, 6, 12, 13, 14	issect 17704	. . . . . 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))))
16	1, 15	mpbid 231	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ (𝑋(Hom ‘𝐷)𝑌) ∧ 𝑁 ∈ (𝑌(Hom ‘𝐷)𝑋) ∧ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀) = ((Id‘𝐷)‘𝑋)))
17	16	simp3d 1144	. . . 4 ⊢ (𝜑 → (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀) = ((Id‘𝐷)‘𝑋))
18	17	fveq2d 6895	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀)) = ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)))
19		eqid 2732	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
20	16	simp1d 1142	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑋(Hom ‘𝐷)𝑌))
21	16	simp2d 1143	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑌(Hom ‘𝐷)𝑋))
22	2, 3, 4, 19, 7, 13, 14, 13, 20, 21	funcco 17825	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)))
23		eqid 2732	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
24	2, 5, 23, 7, 13	funcid 17824	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋)))
25	18, 22, 24	3eqtr3d 2780	. 2 ⊢ (𝜑 → (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐸)‘(𝐹‘𝑋)))
26		eqid 2732	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
27		eqid 2732	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
28		funcsect.t	. . 3 ⊢ 𝑇 = (Sect‘𝐸)
29	11	simprd 496	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
30	2, 26, 7	funcf1 17820	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸))
31	30, 13	ffvelcdmd 7087	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸))
32	30, 14	ffvelcdmd 7087	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸))
33	2, 3, 27, 7, 13, 14	funcf2 17822	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐷)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐸)(𝐹‘𝑌)))
34	33, 20	ffvelcdmd 7087	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)(Hom ‘𝐸)(𝐹‘𝑌)))
35	2, 3, 27, 7, 14, 13	funcf2 17822	. . . 4 ⊢ (𝜑 → (𝑌𝐺𝑋):(𝑌(Hom ‘𝐷)𝑋)⟶((𝐹‘𝑌)(Hom ‘𝐸)(𝐹‘𝑋)))
36	35, 21	ffvelcdmd 7087	. . 3 ⊢ (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹‘𝑌)(Hom ‘𝐸)(𝐹‘𝑋)))
37	26, 27, 19, 23, 28, 29, 31, 32, 34, 36	issect2 17705	. 2 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝑇(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐸)‘(𝐹‘𝑋))))
38	25, 37	mpbird 256	1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝑇(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ⟨cop 4634   class class class wbr 5148  ‘cfv 6543  (class class class)co 7411  Basecbs 17148  Hom chom 17212  compcco 17213  Catccat 17612  Idccid 17613  Sectcsect 17695   Func cfunc 17808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-ixp 8894  df-sect 17698  df-func 17812

This theorem is referenced by:  funcinv  17827