Theorem funcsect 17840

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 2730	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2730	. . . . . . 7 ⊢ (comp‘𝐷) = (comp‘𝐷)
5		eqid 2730	. . . . . . 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 5110	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
9	7, 8	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
10		funcrcl 17831	. . . . . . . . 9 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
11	9, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
12	11	simpld 494	. . . . . . 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 17721	. . . . . 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 232	. . . . 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 6864	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀)) = ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)))
19		eqid 2730	. . . 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 17839	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐷)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)))
23		eqid 2730	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
24	2, 5, 23, 7, 13	funcid 17838	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋)))
25	18, 22, 24	3eqtr3d 2773	. 2 ⊢ (𝜑 → (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐸)‘(𝐹‘𝑋)))
26		eqid 2730	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
27		eqid 2730	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
28		funcsect.t	. . 3 ⊢ 𝑇 = (Sect‘𝐸)
29	11	simprd 495	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
30	2, 26, 7	funcf1 17834	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸))
31	30, 13	ffvelcdmd 7059	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸))
32	30, 14	ffvelcdmd 7059	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸))
33	2, 3, 27, 7, 13, 14	funcf2 17836	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐷)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐸)(𝐹‘𝑌)))
34	33, 20	ffvelcdmd 7059	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)(Hom ‘𝐸)(𝐹‘𝑌)))
35	2, 3, 27, 7, 14, 13	funcf2 17836	. . . 4 ⊢ (𝜑 → (𝑌𝐺𝑋):(𝑌(Hom ‘𝐷)𝑋)⟶((𝐹‘𝑌)(Hom ‘𝐸)(𝐹‘𝑋)))
36	35, 21	ffvelcdmd 7059	. . 3 ⊢ (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹‘𝑌)(Hom ‘𝐸)(𝐹‘𝑋)))
37	26, 27, 19, 23, 28, 29, 31, 32, 34, 36	issect2 17722	. 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 257	1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝑇(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ⟨cop 4597   class class class wbr 5109  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  Hom chom 17237  compcco 17238  Catccat 17631  Idccid 17632  Sectcsect 17712   Func cfunc 17822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-map 8803  df-ixp 8873  df-sect 17715  df-func 17826

This theorem is referenced by:  funcinv  17841