Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fthsect Structured version   Unicode version

Theorem fthsect 14114
 Description: A faithful functor reflects sections. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fthsect.b
fthsect.h
fthsect.f Faith
fthsect.x
fthsect.y
fthsect.m
fthsect.n
fthsect.s Sect
fthsect.t Sect
Assertion
Ref Expression
fthsect

Proof of Theorem fthsect
StepHypRef Expression
1 fthsect.b . . . 4
2 fthsect.h . . . 4
3 eqid 2435 . . . 4
4 fthsect.f . . . 4 Faith
5 fthsect.x . . . 4
6 eqid 2435 . . . . 5 comp comp
7 fthfunc 14096 . . . . . . . . . 10 Faith
87ssbri 4246 . . . . . . . . 9 Faith
94, 8syl 16 . . . . . . . 8
10 df-br 4205 . . . . . . . 8
119, 10sylib 189 . . . . . . 7
12 funcrcl 14052 . . . . . . 7
1311, 12syl 16 . . . . . 6
1413simpld 446 . . . . 5
15 fthsect.y . . . . 5
16 fthsect.m . . . . 5
17 fthsect.n . . . . 5
181, 2, 6, 14, 5, 15, 5, 16, 17catcocl 13902 . . . 4 comp
19 eqid 2435 . . . . 5
201, 2, 19, 14, 5catidcl 13899 . . . 4
211, 2, 3, 4, 5, 5, 18, 20fthi 14107 . . 3 comp comp
22 eqid 2435 . . . . 5 comp comp
231, 2, 6, 22, 9, 5, 15, 5, 16, 17funcco 14060 . . . 4 comp comp
24 eqid 2435 . . . . 5
251, 19, 24, 9, 5funcid 14059 . . . 4
2623, 25eqeq12d 2449 . . 3 comp comp
2721, 26bitr3d 247 . 2 comp comp
28 fthsect.s . . 3 Sect
291, 2, 6, 19, 28, 14, 5, 15, 16, 17issect2 13972 . 2 comp
30 eqid 2435 . . 3
31 fthsect.t . . 3 Sect
3213simprd 450 . . 3
331, 30, 9funcf1 14055 . . . 4
3433, 5ffvelrnd 5863 . . 3
3533, 15ffvelrnd 5863 . . 3
361, 2, 3, 9, 5, 15funcf2 14057 . . . 4
3736, 16ffvelrnd 5863 . . 3
381, 2, 3, 9, 15, 5funcf2 14057 . . . 4
3938, 17ffvelrnd 5863 . . 3
4030, 3, 22, 24, 31, 32, 34, 35, 37, 39issect2 13972 . 2 comp
4127, 29, 403bitr4d 277 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cop 3809   class class class wbr 4204  cfv 5446  (class class class)co 6073  cbs 13461   chom 13532  compcco 13533  ccat 13881  ccid 13882  Sectcsect 13962   cfunc 14043   Faith cfth 14092 This theorem is referenced by:  fthinv  14115 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-map 7012  df-ixp 7056  df-cat 13885  df-cid 13886  df-sect 13965  df-func 14047  df-fth 14094
 Copyright terms: Public domain W3C validator