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Theorem fthpropd 14935
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  |-  ( 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 14923 . 2  |-  Rel  ( A Faith  C )
2 relfth 14923 . 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 14914 . . . . 5  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
1211breqd 4403 . . . 4  |-  ( ph  ->  ( f ( A 
Func  C ) g  <->  f ( B  Func  D ) g ) )
133homfeqbas 14739 . . . . 5  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
1413raleqdv 3021 . . . . 5  |-  ( ph  ->  ( A. y  e.  ( Base `  A
) Fun  `' (
x g y )  <->  A. y  e.  ( Base `  B ) Fun  `' ( x g y ) ) )
1513, 14raleqbidv 3029 . . . 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 710 . . 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 2451 . . . 4  |-  ( Base `  A )  =  (
Base `  A )
1817isfth 14928 . . 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 2451 . . . 4  |-  ( Base `  B )  =  (
Base `  B )
2019isfth 14928 . . 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 5038 1  |-  ( ph  ->  ( A Faith  C )  =  ( B Faith  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   class class class wbr 4392   `'ccnv 4939   Fun wfun 5512   ` cfv 5518  (class class class)co 6192   Basecbs 14278   Hom f chomf 14708  compfccomf 14709    Func cfunc 14868   Faith cfth 14917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-map 7318  df-ixp 7366  df-cat 14710  df-cid 14711  df-homf 14712  df-comf 14713  df-func 14872  df-fth 14919
This theorem is referenced by: (None)
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