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Theorem fullpropd 14072
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
fullpropd  |-  ( ph  ->  ( A Full  C )  =  ( B Full  D
) )

Proof of Theorem fullpropd
Dummy variables  f 
g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfull 14060 . 2  |-  Rel  ( A Full  C )
2 relfull 14060 . 2  |-  Rel  ( B Full  D )
3 fullpropd.1 . . . . . . . 8  |-  ( ph  ->  (  Homf 
`  A )  =  (  Homf 
`  B ) )
43homfeqbas 13877 . . . . . . 7  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
54adantr 452 . . . . . 6  |-  ( (
ph  /\  f ( A  Func  C ) g )  ->  ( Base `  A )  =  (
Base `  B )
)
65adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  f
( A  Func  C
) g )  /\  x  e.  ( Base `  A ) )  -> 
( Base `  A )  =  ( Base `  B
) )
7 eqid 2404 . . . . . . . . 9  |-  ( Base `  C )  =  (
Base `  C )
8 eqid 2404 . . . . . . . . 9  |-  (  Hom  `  C )  =  (  Hom  `  C )
9 eqid 2404 . . . . . . . . 9  |-  (  Hom  `  D )  =  (  Hom  `  D )
10 fullpropd.3 . . . . . . . . . 10  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
1110ad3antrrr 711 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
(  Homf 
`  C )  =  (  Homf 
`  D ) )
12 eqid 2404 . . . . . . . . . . 11  |-  ( Base `  A )  =  (
Base `  A )
13 simpllr 736 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
f ( A  Func  C ) g )
1412, 7, 13funcf1 14018 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
f : ( Base `  A ) --> ( Base `  C ) )
15 simplr 732 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  ->  x  e.  ( Base `  A ) )
1614, 15ffvelrnd 5830 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( f `  x
)  e.  ( Base `  C ) )
17 simpr 448 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
y  e.  ( Base `  A ) )
1814, 17ffvelrnd 5830 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( f `  y
)  e.  ( Base `  C ) )
197, 8, 9, 11, 16, 18homfeqval 13878 . . . . . . . 8  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( ( f `  x ) (  Hom  `  C ) ( f `
 y ) )  =  ( ( f `
 x ) (  Hom  `  D )
( f `  y
) ) )
2019eqeq2d 2415 . . . . . . 7  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  C ) ( f `
 y ) )  <->  ran  ( x g y )  =  ( ( f `  x ) (  Hom  `  D
) ( f `  y ) ) ) )
216, 20raleqbidva 2878 . . . . . 6  |-  ( ( ( ph  /\  f
( A  Func  C
) g )  /\  x  e.  ( Base `  A ) )  -> 
( A. y  e.  ( Base `  A
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  C ) ( f `
 y ) )  <->  A. y  e.  ( Base `  B ) ran  ( x g y )  =  ( ( f `  x ) (  Hom  `  D
) ( f `  y ) ) ) )
225, 21raleqbidva 2878 . . . . 5  |-  ( (
ph  /\  f ( A  Func  C ) g )  ->  ( A. x  e.  ( Base `  A ) A. y  e.  ( Base `  A
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  C ) ( f `
 y ) )  <->  A. x  e.  ( Base `  B ) A. y  e.  ( Base `  B ) ran  (
x g y )  =  ( ( f `
 x ) (  Hom  `  D )
( f `  y
) ) ) )
2322pm5.32da 623 . . . 4  |-  ( ph  ->  ( ( f ( A  Func  C )
g  /\  A. x  e.  ( Base `  A
) A. y  e.  ( Base `  A
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  C ) ( f `
 y ) ) )  <->  ( f ( A  Func  C )
g  /\  A. x  e.  ( Base `  B
) A. y  e.  ( Base `  B
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  D ) ( f `
 y ) ) ) ) )
24 fullpropd.2 . . . . . . 7  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
25 fullpropd.4 . . . . . . 7  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
26 fullpropd.a . . . . . . 7  |-  ( ph  ->  A  e.  V )
27 fullpropd.b . . . . . . 7  |-  ( ph  ->  B  e.  V )
28 fullpropd.c . . . . . . 7  |-  ( ph  ->  C  e.  V )
29 fullpropd.d . . . . . . 7  |-  ( ph  ->  D  e.  V )
303, 24, 10, 25, 26, 27, 28, 29funcpropd 14052 . . . . . 6  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
3130breqd 4183 . . . . 5  |-  ( ph  ->  ( f ( A 
Func  C ) g  <->  f ( B  Func  D ) g ) )
3231anbi1d 686 . . . 4  |-  ( ph  ->  ( ( f ( A  Func  C )
g  /\  A. x  e.  ( Base `  B
) A. y  e.  ( Base `  B
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  D ) ( f `
 y ) ) )  <->  ( f ( B  Func  D )
g  /\  A. x  e.  ( Base `  B
) A. y  e.  ( Base `  B
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  D ) ( f `
 y ) ) ) ) )
3323, 32bitrd 245 . . 3  |-  ( ph  ->  ( ( f ( A  Func  C )
g  /\  A. x  e.  ( Base `  A
) A. y  e.  ( Base `  A
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  C ) ( f `
 y ) ) )  <->  ( f ( B  Func  D )
g  /\  A. x  e.  ( Base `  B
) A. y  e.  ( Base `  B
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  D ) ( f `
 y ) ) ) ) )
3412, 8isfull 14062 . . 3  |-  ( f ( A Full  C ) g  <->  ( f ( A  Func  C )
g  /\  A. x  e.  ( Base `  A
) A. y  e.  ( Base `  A
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  C ) ( f `
 y ) ) ) )
35 eqid 2404 . . . 4  |-  ( Base `  B )  =  (
Base `  B )
3635, 9isfull 14062 . . 3  |-  ( f ( B Full  D ) g  <->  ( f ( B  Func  D )
g  /\  A. x  e.  ( Base `  B
) A. y  e.  ( Base `  B
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  D ) ( f `
 y ) ) ) )
3733, 34, 363bitr4g 280 . 2  |-  ( ph  ->  ( f ( A Full 
C ) g  <->  f ( B Full  D ) g ) )
381, 2, 37eqbrrdiv 4933 1  |-  ( ph  ->  ( A Full  C )  =  ( B Full  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   class class class wbr 4172   ran crn 4838   ` cfv 5413  (class class class)co 6040   Basecbs 13424    Hom chom 13495    Homf chomf 13846  compfccomf 13847    Func cfunc 14006   Full cful 14054
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-map 6979  df-ixp 7023  df-cat 13848  df-cid 13849  df-homf 13850  df-comf 13851  df-func 14010  df-full 14056
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