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Theorem fthpropd 14077
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 14065 . 2  |-  Rel  ( A Faith  C )
2 relfth 14065 . 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 14056 . . . . 5  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
1211breqd 4187 . . . 4  |-  ( ph  ->  ( f ( A 
Func  C ) g  <->  f ( B  Func  D ) g ) )
133homfeqbas 13881 . . . . 5  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
1413raleqdv 2874 . . . . 5  |-  ( ph  ->  ( A. y  e.  ( Base `  A
) Fun  `' (
x g y )  <->  A. y  e.  ( Base `  B ) Fun  `' ( x g y ) ) )
1513, 14raleqbidv 2880 . . . 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 692 . . 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 2408 . . . 4  |-  ( Base `  A )  =  (
Base `  A )
1817isfth 14070 . . 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 2408 . . . 4  |-  ( Base `  B )  =  (
Base `  B )
2019isfth 14070 . . 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 280 . 2  |-  ( ph  ->  ( f ( A Faith 
C ) g  <->  f ( B Faith  D ) g ) )
221, 2, 21eqbrrdiv 4937 1  |-  ( ph  ->  ( A Faith  C )  =  ( B Faith  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   class class class wbr 4176   `'ccnv 4840   Fun wfun 5411   ` cfv 5417  (class class class)co 6044   Basecbs 13428    Homf chomf 13850  compfccomf 13851    Func cfunc 14010   Faith cfth 14059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-map 6983  df-ixp 7027  df-cat 13852  df-cid 13853  df-homf 13854  df-comf 13855  df-func 14014  df-fth 14061
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