Theorem fthsect 17826

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 2730	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		fthsect.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
5		fthsect.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		eqid 2730	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		fthfunc 17808	. . . . . . . . . 10 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
8	7	ssbri 5134	. . . . . . . . 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 5090	. . . . . . . 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 17762	. . . . . . 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 17583	. . . 4 ⊢ (𝜑 → (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) ∈ (𝑋𝐻𝑋))
19		eqid 2730	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
20	1, 2, 19, 14, 5	catidcl 17580	. . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
21	1, 2, 3, 4, 5, 5, 18, 20	fthi 17819	. . 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 2730	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
23	1, 2, 6, 22, 9, 5, 15, 5, 16, 17	funcco 17770	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘(𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀)) = (((𝑌𝐺𝑋)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)))
24		eqid 2730	. . . . 5 ⊢ (Id‘𝐷) = (Id‘𝐷)
25	1, 19, 24, 9, 5	funcid 17769	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐶)‘𝑋)) = ((Id‘𝐷)‘(𝐹‘𝑋)))
26	23, 25	eqeq12d 2746	. . 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 17653	. 2 ⊢ (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ (𝑁(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑀) = ((Id‘𝐶)‘𝑋)))
30		eqid 2730	. . 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 17765	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷))
34	33, 5	ffvelcdmd 7013	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷))
35	33, 15	ffvelcdmd 7013	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷))
36	1, 2, 3, 9, 5, 15	funcf2 17767	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
37	36, 16	ffvelcdmd 7013	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
38	1, 2, 3, 9, 15, 5	funcf2 17767	. . . 4 ⊢ (𝜑 → (𝑌𝐺𝑋):(𝑌𝐻𝑋)⟶((𝐹‘𝑌)(Hom ‘𝐷)(𝐹‘𝑋)))
39	38, 17	ffvelcdmd 7013	. . 3 ⊢ (𝜑 → ((𝑌𝐺𝑋)‘𝑁) ∈ ((𝐹‘𝑌)(Hom ‘𝐷)(𝐹‘𝑋)))
40	30, 3, 22, 24, 31, 32, 34, 35, 37, 39	issect2 17653	. 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 2110  ⟨cop 4580   class class class wbr 5089  ‘cfv 6477  (class class class)co 7341  Basecbs 17112  Hom chom 17164  compcco 17165  Catccat 17562  Idccid 17563  Sectcsect 17643   Func cfunc 17753   Faith cfth 17804

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

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 2067  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-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-map 8747  df-ixp 8817  df-cat 17566  df-cid 17567  df-sect 17646  df-func 17757  df-fth 17806

This theorem is referenced by:  fthinv  17827