Theorem funcsect 17818

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 5148	. . . . . . . . . 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 17809	. . . . . . . . 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 17696	. . . . . 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 6892	. . 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 17817	. . 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 17816	. . 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 17812	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸))
31	30, 13	ffvelcdmd 7084	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸))
32	30, 14	ffvelcdmd 7084	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸))
33	2, 3, 27, 7, 13, 14	funcf2 17814	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐷)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐸)(𝐹‘𝑌)))
34	33, 20	ffvelcdmd 7084	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)(Hom ‘𝐸)(𝐹‘𝑌)))
35	2, 3, 27, 7, 14, 13	funcf2 17814	. . . 4 ⊢ (𝜑 → (𝑌𝐺𝑋):(𝑌(Hom ‘𝐷)𝑋)⟶((𝐹‘𝑌)(Hom ‘𝐸)(𝐹‘𝑋)))
36	35, 21	ffvelcdmd 7084	. . 3 ⊢ (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹‘𝑌)(Hom ‘𝐸)(𝐹‘𝑋)))
37	26, 27, 19, 23, 28, 29, 31, 32, 34, 36	issect2 17697	. 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 4633   class class class wbr 5147  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  Hom chom 17204  compcco 17205  Catccat 17604  Idccid 17605  Sectcsect 17687   Func cfunc 17800

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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-ixp 8888  df-sect 17690  df-func 17804

This theorem is referenced by:  funcinv  17819