Theorem fthsect 17972

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 2765	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		fthsect.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
5		fthsect.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		eqid 2765	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		fthfunc 17954	. . . . . . . . . 10 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
8	7	ssbri 5149	. . . . . . . . 9 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
9	4, 8	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
10		df-br 5105	. . . . . . . 8 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
11	9, 10	sylib 221	. . . . . . 7 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12		funcrcl 17908	. . . . . . 7 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
13	11, 12	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
14	13	simpld 499	. . . . 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 17729	. . . 4 ⊢ (𝜑 → (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) ∈ (𝑋𝐻𝑋))
19		eqid 2765	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
20	1, 2, 19, 14, 5	catidcl 17726	. . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
21	1, 2, 3, 4, 5, 5, 18, 20	fthi 17965	. . 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 2765	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
23	1, 2, 6, 22, 9, 5, 15, 5, 16, 17	funcco 17916	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)))
24		eqid 2765	. . . . 5 ⊢ (Id‘𝐷) = (Id‘𝐷)
25	1, 19, 24, 9, 5	funcid 17915	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐶)‘𝑋)) = ((Id‘𝐷)‘(𝐹‘𝑋)))
26	23, 25	eqeq12d 2781	. . 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 284	. 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 17799	. 2 ⊢ (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) = ((Id‘𝐶)‘𝑋)))
30		eqid 2765	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
31		fthsect.t	. . 3 ⊢ 𝑇 = (Sect‘𝐷)
32	13	simprd 500	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
33	1, 30, 9	funcf1 17911	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷))
34	33, 5	ffvelcdmd 7070	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷))
35	33, 15	ffvelcdmd 7070	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷))
36	1, 2, 3, 9, 5, 15	funcf2 17913	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
37	36, 16	ffvelcdmd 7070	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
38	1, 2, 3, 9, 15, 5	funcf2 17913	. . . 4 ⊢ (𝜑 → (𝑌𝐺𝑋):(𝑌𝐻𝑋)⟶((𝐹‘𝑌)(Hom ‘𝐷)(𝐹‘𝑋)))
39	38, 17	ffvelcdmd 7070	. . 3 ⊢ (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹‘𝑌)(Hom ‘𝐷)(𝐹‘𝑋)))
40	30, 3, 22, 24, 31, 32, 34, 35, 37, 39	issect2 17799	. 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 314	1 ⊢ (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝑇(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ⟨cop 4591   class class class wbr 5104  ‘cfv 6525  (class class class)co 7400  Basecbs 17257  Hom chom 17309  compcco 17310  Catccat 17708  Idccid 17709  Sectcsect 17789   Func cfunc 17899   Faith cfth 17950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-ixp 8884  df-cat 17712  df-cid 17713  df-sect 17792  df-func 17903  df-fth 17952

This theorem is referenced by:  fthinv  17973