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Theorem fthpropd 15534
 Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same faithful functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fullpropd.1 f f
fullpropd.2 compf compf
fullpropd.3 f f
fullpropd.4 compf compf
fullpropd.a
fullpropd.b
fullpropd.c
fullpropd.d
Assertion
Ref Expression
fthpropd Faith Faith

Proof of Theorem fthpropd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfth 15522 . 2 Faith
2 relfth 15522 . 2 Faith
3 fullpropd.1 . . . . . 6 f f
4 fullpropd.2 . . . . . 6 compf compf
5 fullpropd.3 . . . . . 6 f f
6 fullpropd.4 . . . . . 6 compf compf
7 fullpropd.a . . . . . 6
8 fullpropd.b . . . . . 6
9 fullpropd.c . . . . . 6
10 fullpropd.d . . . . . 6
113, 4, 5, 6, 7, 8, 9, 10funcpropd 15513 . . . . 5
1211breqd 4406 . . . 4
133homfeqbas 15309 . . . . 5
1413raleqdv 3010 . . . . 5
1513, 14raleqbidv 3018 . . . 4
1612, 15anbi12d 709 . . 3
17 eqid 2402 . . . 4
1817isfth 15527 . . 3 Faith
19 eqid 2402 . . . 4
2019isfth 15527 . . 3 Faith
2116, 18, 203bitr4g 288 . 2 Faith Faith
221, 2, 21eqbrrdiv 4922 1 Faith Faith
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   wceq 1405   wcel 1842  wral 2754   class class class wbr 4395  ccnv 4822   wfun 5563  cfv 5569  (class class class)co 6278  cbs 14841   f chomf 15280  compfccomf 15281   cfunc 15467   Faith cfth 15516 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-map 7459  df-ixp 7508  df-cat 15282  df-cid 15283  df-homf 15284  df-comf 15285  df-func 15471  df-fth 15518 This theorem is referenced by: (None)
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