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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  |-  ( ph  ->  (  Homf 
`  A )  =  (  Homf 
`  B ) )
fullpropd.2  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
fullpropd.3  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  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 14111 . 2  |-  Rel  ( A Faith  C )
2 relfth 14111 . 2  |-  Rel  ( B Faith  D )
3 fullpropd.1 . . . . . 6  |-  ( ph  ->  (  Homf 
`  A )  =  (  Homf 
`  B ) )
4 fullpropd.2 . . . . . 6  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
5 fullpropd.3 . . . . . 6  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  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 14102 . . . . 5  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
1211breqd 4226 . . . 4  |-  ( ph  ->  ( f ( A 
Func  C ) g  <->  f ( B  Func  D ) g ) )
133homfeqbas 13927 . . . . 5  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
1413raleqdv 2912 . . . . 5  |-  ( ph  ->  ( A. y  e.  ( Base `  A
) Fun  `' (
x g y )  <->  A. y  e.  ( Base `  B ) Fun  `' ( x g y ) ) )
1513, 14raleqbidv 2918 . . . 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 693 . . 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 2438 . . . 4  |-  ( Base `  A )  =  (
Base `  A )
1817isfth 14116 . . 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 2438 . . . 4  |-  ( Base `  B )  =  (
Base `  B )
2019isfth 14116 . . 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 281 . 2  |-  ( ph  ->  ( f ( A Faith 
C ) g  <->  f ( B Faith  D ) g ) )
221, 2, 21eqbrrdiv 4977 1  |-  ( ph  ->  ( A Faith  C )  =  ( B Faith  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   class class class wbr 4215   `'ccnv 4880   Fun wfun 5451   ` cfv 5457  (class class class)co 6084   Basecbs 13474    Homf chomf 13896  compfccomf 13897    Func 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
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