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Theorem fthsect 15781
 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 2429 . . . 4
4 fthsect.f . . . 4 Faith
5 fthsect.x . . . 4
6 eqid 2429 . . . . 5 comp comp
7 fthfunc 15763 . . . . . . . . . 10 Faith
87ssbri 4468 . . . . . . . . 9 Faith
94, 8syl 17 . . . . . . . 8
10 df-br 4427 . . . . . . . 8
119, 10sylib 199 . . . . . . 7
12 funcrcl 15719 . . . . . . 7
1311, 12syl 17 . . . . . 6
1413simpld 460 . . . . 5
15 fthsect.y . . . . 5
16 fthsect.m . . . . 5
17 fthsect.n . . . . 5
181, 2, 6, 14, 5, 15, 5, 16, 17catcocl 15542 . . . 4 comp
19 eqid 2429 . . . . 5
201, 2, 19, 14, 5catidcl 15539 . . . 4
211, 2, 3, 4, 5, 5, 18, 20fthi 15774 . . 3 comp comp
22 eqid 2429 . . . . 5 comp comp
231, 2, 6, 22, 9, 5, 15, 5, 16, 17funcco 15727 . . . 4 comp comp
24 eqid 2429 . . . . 5
251, 19, 24, 9, 5funcid 15726 . . . 4
2623, 25eqeq12d 2451 . . 3 comp comp
2721, 26bitr3d 258 . 2 comp comp
28 fthsect.s . . 3 Sect
291, 2, 6, 19, 28, 14, 5, 15, 16, 17issect2 15610 . 2 comp
30 eqid 2429 . . 3
31 fthsect.t . . 3 Sect
3213simprd 464 . . 3
331, 30, 9funcf1 15722 . . . 4
3433, 5ffvelrnd 6038 . . 3
3533, 15ffvelrnd 6038 . . 3
361, 2, 3, 9, 5, 15funcf2 15724 . . . 4
3736, 16ffvelrnd 6038 . . 3
381, 2, 3, 9, 15, 5funcf2 15724 . . . 4
3938, 17ffvelrnd 6038 . . 3
4030, 3, 22, 24, 31, 32, 34, 35, 37, 39issect2 15610 . 2 comp
4127, 29, 403bitr4d 288 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1870  cop 4008   class class class wbr 4426  cfv 5601  (class class class)co 6305  cbs 15084   chom 15163  compcco 15164  ccat 15521  ccid 15522  Sectcsect 15600   cfunc 15710   Faith cfth 15759 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482  df-ixp 7531  df-cat 15525  df-cid 15526  df-sect 15603  df-func 15714  df-fth 15761 This theorem is referenced by:  fthinv  15782
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