Theorem fthsect 17872

Description: A faithful functor reflects sections. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
fthsect.b	⊢ 𝐵 = (Base‘𝐶)
fthsect.h	⊢ 𝐻 = (Hom ‘𝐶)
fthsect.f	⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
fthsect.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
fthsect.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
fthsect.m	⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))
fthsect.n	⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑋))
fthsect.s	⊢ 𝑆 = (Sect‘𝐶)
fthsect.t	⊢ 𝑇 = (Sect‘𝐷)

Assertion

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

Proof of Theorem fthsect

Step	Hyp	Ref	Expression
1		fthsect.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		fthsect.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
3		eqid 2732	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		fthsect.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
5		fthsect.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		eqid 2732	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		fthfunc 17854	. . . . . . . . . 10 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
8	7	ssbri 5192	. . . . . . . . 9 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
9	4, 8	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
10		df-br 5148	. . . . . . . 8 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
11	9, 10	sylib 217	. . . . . . 7 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12		funcrcl 17809	. . . . . . 7 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
13	11, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
14	13	simpld 495	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
15		fthsect.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
16		fthsect.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))
17		fthsect.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑋))
18	1, 2, 6, 14, 5, 15, 5, 16, 17	catcocl 17625	. . . 4 ⊢ (𝜑 → (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) ∈ (𝑋𝐻𝑋))
19		eqid 2732	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
20	1, 2, 19, 14, 5	catidcl 17622	. . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
21	1, 2, 3, 4, 5, 5, 18, 20	fthi 17865	. . 3 ⊢ (𝜑 → (((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀)) = ((𝑋𝐺𝑋)‘((Id‘𝐶)‘𝑋)) ↔ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) = ((Id‘𝐶)‘𝑋)))
22		eqid 2732	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
23	1, 2, 6, 22, 9, 5, 15, 5, 16, 17	funcco 17817	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)))
24		eqid 2732	. . . . 5 ⊢ (Id‘𝐷) = (Id‘𝐷)
25	1, 19, 24, 9, 5	funcid 17816	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐶)‘𝑋)) = ((Id‘𝐷)‘(𝐹‘𝑋)))
26	23, 25	eqeq12d 2748	. . 3 ⊢ (𝜑 → (((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀)) = ((𝑋𝐺𝑋)‘((Id‘𝐶)‘𝑋)) ↔ (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐷)‘(𝐹‘𝑋))))
27	21, 26	bitr3d 280	. 2 ⊢ (𝜑 → ((𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) = ((Id‘𝐶)‘𝑋) ↔ (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐷)‘(𝐹‘𝑋))))
28		fthsect.s	. . 3 ⊢ 𝑆 = (Sect‘𝐶)
29	1, 2, 6, 19, 28, 14, 5, 15, 16, 17	issect2 17697	. 2 ⊢ (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) = ((Id‘𝐶)‘𝑋)))
30		eqid 2732	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
31		fthsect.t	. . 3 ⊢ 𝑇 = (Sect‘𝐷)
32	13	simprd 496	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
33	1, 30, 9	funcf1 17812	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷))
34	33, 5	ffvelcdmd 7084	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷))
35	33, 15	ffvelcdmd 7084	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷))
36	1, 2, 3, 9, 5, 15	funcf2 17814	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
37	36, 16	ffvelcdmd 7084	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
38	1, 2, 3, 9, 15, 5	funcf2 17814	. . . 4 ⊢ (𝜑 → (𝑌𝐺𝑋):(𝑌𝐻𝑋)⟶((𝐹‘𝑌)(Hom ‘𝐷)(𝐹‘𝑋)))
39	38, 17	ffvelcdmd 7084	. . 3 ⊢ (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹‘𝑌)(Hom ‘𝐷)(𝐹‘𝑋)))
40	30, 3, 22, 24, 31, 32, 34, 35, 37, 39	issect2 17697	. 2 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝑇(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)) = ((Id‘𝐷)‘(𝐹‘𝑋))))
41	27, 29, 40	3bitr4d 310	1 ⊢ (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝑇(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = 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   Faith cfth 17850

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-rmo 3376  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-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-ixp 8888  df-cat 17608  df-cid 17609  df-sect 17690  df-func 17804  df-fth 17852

This theorem is referenced by:  fthinv  17873