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Theorem fullpropd 15336
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
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 15324 . 2  |-  Rel  ( A Full  C )
2 relfull 15324 . 2  |-  Rel  ( B Full  D )
3 fullpropd.1 . . . . . . . 8  |-  ( ph  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
43homfeqbas 15112 . . . . . . 7  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
54adantr 465 . . . . . 6  |-  ( (
ph  /\  f ( A  Func  C ) g )  ->  ( Base `  A )  =  (
Base `  B )
)
65adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  f
( A  Func  C
) g )  /\  x  e.  ( Base `  A ) )  -> 
( Base `  A )  =  ( Base `  B
) )
7 eqid 2457 . . . . . . . . 9  |-  ( Base `  C )  =  (
Base `  C )
8 eqid 2457 . . . . . . . . 9  |-  ( Hom  `  C )  =  ( Hom  `  C )
9 eqid 2457 . . . . . . . . 9  |-  ( Hom  `  D )  =  ( Hom  `  D )
10 fullpropd.3 . . . . . . . . . 10  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
1110ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( Hom f  `  C )  =  ( Hom f  `  D ) )
12 eqid 2457 . . . . . . . . . . 11  |-  ( Base `  A )  =  (
Base `  A )
13 simpllr 760 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
f ( A  Func  C ) g )
1412, 7, 13funcf1 15282 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
f : ( Base `  A ) --> ( Base `  C ) )
15 simplr 755 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  ->  x  e.  ( Base `  A ) )
1614, 15ffvelrnd 6033 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( f `  x
)  e.  ( Base `  C ) )
17 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
y  e.  ( Base `  A ) )
1814, 17ffvelrnd 6033 . . . . . . . . 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 15113 . . . . . . . 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 2471 . . . . . . 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 3070 . . . . . 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 3070 . . . . 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 641 . . . 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 15316 . . . . . 6  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
3130breqd 4467 . . . . 5  |-  ( ph  ->  ( f ( A 
Func  C ) g  <->  f ( B  Func  D ) g ) )
3231anbi1d 704 . . . 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 253 . . 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 15326 . . 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 2457 . . . 4  |-  ( Base `  B )  =  (
Base `  B )
3635, 9isfull 15326 . . 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 288 . 2  |-  ( ph  ->  ( f ( A Full 
C ) g  <->  f ( B Full  D ) g ) )
381, 2, 37eqbrrdiv 5110 1  |-  ( ph  ->  ( A Full  C )  =  ( B Full  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   class class class wbr 4456   ran crn 5009   ` cfv 5594  (class class class)co 6296   Basecbs 14644   Hom chom 14723   Hom f chomf 15083  compfccomf 15084    Func cfunc 15270   Full cful 15318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-map 7440  df-ixp 7489  df-cat 15085  df-cid 15086  df-homf 15087  df-comf 15088  df-func 15274  df-full 15320
This theorem is referenced by: (None)
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