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Theorem isffth2 13790
 Description: A fully faithful functor is a functor which is bijective on hom-sets. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
isfth.b
isfth.h
isfth.j
Assertion
Ref Expression
isffth2 Full Faith
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,   ,,   ,,

Proof of Theorem isffth2
StepHypRef Expression
1 isfth.b . . . 4
2 isfth.j . . . 4
3 isfth.h . . . 4
41, 2, 3isfull2 13785 . . 3 Full
51, 3, 2isfth2 13789 . . 3 Faith
64, 5anbi12i 678 . 2 Full Faith
7 brin 4070 . 2 Full Faith Full Faith
8 df-f1o 5262 . . . . . . . 8
9 ancom 437 . . . . . . . 8
108, 9bitri 240 . . . . . . 7
1110ralbii 2567 . . . . . 6
1211ralbii 2567 . . . . 5
13 r19.26-2 2676 . . . . 5
1412, 13bitri 240 . . . 4
1514anbi2i 675 . . 3
16 anandi 801 . . 3
1715, 16bitri 240 . 2
186, 7, 173bitr4i 268 1 Full Faith
 Colors of variables: wff set class Syntax hints:   wb 176   wa 358   wceq 1623  wral 2543   cin 3151   class class class wbr 4023  wf1 5252  wfo 5253  wf1o 5254  cfv 5255  (class class class)co 5858  cbs 13148   chom 13219   cfunc 13728   Full cful 13776   Faith cfth 13777 This theorem is referenced by:  idffth  13807  ressffth  13812  catciso  13939  yonffthlem  14056 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-ixp 6818  df-func 13732  df-full 13778  df-fth 13779
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