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

Theorem fthpropd 14123
 Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same full 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 14111 . 2 Faith
2 relfth 14111 . 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 14102 . . . . 5
1211breqd 4226 . . . 4
133homfeqbas 13927 . . . . 5
1413raleqdv 2912 . . . . 5
1513, 14raleqbidv 2918 . . . 4
1612, 15anbi12d 693 . . 3
17 eqid 2438 . . . 4
1817isfth 14116 . . 3 Faith
19 eqid 2438 . . . 4
2019isfth 14116 . . 3 Faith
2116, 18, 203bitr4g 281 . 2 Faith Faith
221, 2, 21eqbrrdiv 4977 1 Faith Faith
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  wral 2707   class class class wbr 4215  ccnv 4880   wfun 5451  cfv 5457  (class class class)co 6084  cbs 13474   f chomf 13896  compfccomf 13897   cfunc 14056   Faith cfth 14105 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-map 7023  df-ixp 7067  df-cat 13898  df-cid 13899  df-homf 13900  df-comf 13901  df-func 14060  df-fth 14107
 Copyright terms: Public domain W3C validator