Theorem fthsect 17840

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 2731	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		fthsect.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
5		fthsect.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		eqid 2731	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		fthfunc 17822	. . . . . . . . . 10 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
8	7	ssbri 5138	. . . . . . . . 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 5094	. . . . . . . 8 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
11	9, 10	sylib 218	. . . . . . 7 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12		funcrcl 17776	. . . . . . 7 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
13	11, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
14	13	simpld 494	. . . . 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 17597	. . . 4 ⊢ (𝜑 → (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) ∈ (𝑋𝐻𝑋))
19		eqid 2731	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
20	1, 2, 19, 14, 5	catidcl 17594	. . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
21	1, 2, 3, 4, 5, 5, 18, 20	fthi 17833	. . 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 2731	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
23	1, 2, 6, 22, 9, 5, 15, 5, 16, 17	funcco 17784	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)))
24		eqid 2731	. . . . 5 ⊢ (Id‘𝐷) = (Id‘𝐷)
25	1, 19, 24, 9, 5	funcid 17783	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐶)‘𝑋)) = ((Id‘𝐷)‘(𝐹‘𝑋)))
26	23, 25	eqeq12d 2747	. . 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 281	. 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 17667	. 2 ⊢ (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) = ((Id‘𝐶)‘𝑋)))
30		eqid 2731	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
31		fthsect.t	. . 3 ⊢ 𝑇 = (Sect‘𝐷)
32	13	simprd 495	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
33	1, 30, 9	funcf1 17779	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷))
34	33, 5	ffvelcdmd 7024	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷))
35	33, 15	ffvelcdmd 7024	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷))
36	1, 2, 3, 9, 5, 15	funcf2 17781	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
37	36, 16	ffvelcdmd 7024	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
38	1, 2, 3, 9, 15, 5	funcf2 17781	. . . 4 ⊢ (𝜑 → (𝑌𝐺𝑋):(𝑌𝐻𝑋)⟶((𝐹‘𝑌)(Hom ‘𝐷)(𝐹‘𝑋)))
39	38, 17	ffvelcdmd 7024	. . 3 ⊢ (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹‘𝑌)(Hom ‘𝐷)(𝐹‘𝑋)))
40	30, 3, 22, 24, 31, 32, 34, 35, 37, 39	issect2 17667	. 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 311	1 ⊢ (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝑇(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⟨cop 4581   class class class wbr 5093  ‘cfv 6487  (class class class)co 7352  Basecbs 17126  Hom chom 17178  compcco 17179  Catccat 17576  Idccid 17577  Sectcsect 17657   Func cfunc 17767   Faith cfth 17818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-map 8758  df-ixp 8828  df-cat 17580  df-cid 17581  df-sect 17660  df-func 17771  df-fth 17820

This theorem is referenced by:  fthinv  17841