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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  |-  ( ph  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
fullpropd.2  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
fullpropd.3  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
fullpropd.4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
fullpropd.a  |-  ( ph  ->  A  e.  V )
fullpropd.b  |-  ( ph  ->  B  e.  V )
fullpropd.c  |-  ( ph  ->  C  e.  V )
fullpropd.d  |-  ( ph  ->  D  e.  V )
Assertion
Ref Expression
fthpropd  |-  ( ph  ->  ( A Faith  C )  =  ( B Faith  D
) )

Proof of Theorem fthpropd
Dummy variables  f 
g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfth 15522 . 2  |-  Rel  ( A Faith  C )
2 relfth 15522 . 2  |-  Rel  ( B Faith  D )
3 fullpropd.1 . . . . . 6  |-  ( ph  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
4 fullpropd.2 . . . . . 6  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
5 fullpropd.3 . . . . . 6  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
6 fullpropd.4 . . . . . 6  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
7 fullpropd.a . . . . . 6  |-  ( ph  ->  A  e.  V )
8 fullpropd.b . . . . . 6  |-  ( ph  ->  B  e.  V )
9 fullpropd.c . . . . . 6  |-  ( ph  ->  C  e.  V )
10 fullpropd.d . . . . . 6  |-  ( ph  ->  D  e.  V )
113, 4, 5, 6, 7, 8, 9, 10funcpropd 15513 . . . . 5  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
1211breqd 4406 . . . 4  |-  ( ph  ->  ( f ( A 
Func  C ) g  <->  f ( B  Func  D ) g ) )
133homfeqbas 15309 . . . . 5  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
1413raleqdv 3010 . . . . 5  |-  ( ph  ->  ( A. y  e.  ( Base `  A
) Fun  `' (
x g y )  <->  A. y  e.  ( Base `  B ) Fun  `' ( x g y ) ) )
1513, 14raleqbidv 3018 . . . 4  |-  ( ph  ->  ( A. x  e.  ( Base `  A
) A. y  e.  ( Base `  A
) Fun  `' (
x g y )  <->  A. x  e.  ( Base `  B ) A. y  e.  ( Base `  B ) Fun  `' ( x g y ) ) )
1612, 15anbi12d 709 . . 3  |-  ( ph  ->  ( ( f ( A  Func  C )
g  /\  A. x  e.  ( Base `  A
) A. y  e.  ( Base `  A
) Fun  `' (
x g y ) )  <->  ( f ( B  Func  D )
g  /\  A. x  e.  ( Base `  B
) A. y  e.  ( Base `  B
) Fun  `' (
x g y ) ) ) )
17 eqid 2402 . . . 4  |-  ( Base `  A )  =  (
Base `  A )
1817isfth 15527 . . 3  |-  ( f ( A Faith  C ) g  <->  ( f ( A  Func  C )
g  /\  A. x  e.  ( Base `  A
) A. y  e.  ( Base `  A
) Fun  `' (
x g y ) ) )
19 eqid 2402 . . . 4  |-  ( Base `  B )  =  (
Base `  B )
2019isfth 15527 . . 3  |-  ( f ( B Faith  D ) g  <->  ( f ( B  Func  D )
g  /\  A. x  e.  ( Base `  B
) A. y  e.  ( Base `  B
) Fun  `' (
x g y ) ) )
2116, 18, 203bitr4g 288 . 2  |-  ( ph  ->  ( f ( A Faith 
C ) g  <->  f ( B Faith  D ) g ) )
221, 2, 21eqbrrdiv 4922 1  |-  ( ph  ->  ( A Faith  C )  =  ( B Faith  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   class class class wbr 4395   `'ccnv 4822   Fun wfun 5563   ` cfv 5569  (class class class)co 6278   Basecbs 14841   Hom f chomf 15280  compfccomf 15281    Func 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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