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Theorem fullpropd 14826
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 14814 . 2  |-  Rel  ( A Full  C )
2 relfull 14814 . 2  |-  Rel  ( B Full  D )
3 fullpropd.1 . . . . . . . 8  |-  ( ph  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
43homfeqbas 14631 . . . . . . 7  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
54adantr 462 . . . . . 6  |-  ( (
ph  /\  f ( A  Func  C ) g )  ->  ( Base `  A )  =  (
Base `  B )
)
65adantr 462 . . . . . . 7  |-  ( ( ( ph  /\  f
( A  Func  C
) g )  /\  x  e.  ( Base `  A ) )  -> 
( Base `  A )  =  ( Base `  B
) )
7 eqid 2441 . . . . . . . . 9  |-  ( Base `  C )  =  (
Base `  C )
8 eqid 2441 . . . . . . . . 9  |-  ( Hom  `  C )  =  ( Hom  `  C )
9 eqid 2441 . . . . . . . . 9  |-  ( Hom  `  D )  =  ( Hom  `  D )
10 fullpropd.3 . . . . . . . . . 10  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
1110ad3antrrr 724 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( Hom f  `  C )  =  ( Hom f  `  D ) )
12 eqid 2441 . . . . . . . . . . 11  |-  ( Base `  A )  =  (
Base `  A )
13 simpllr 753 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
f ( A  Func  C ) g )
1412, 7, 13funcf1 14772 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
f : ( Base `  A ) --> ( Base `  C ) )
15 simplr 749 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  ->  x  e.  ( Base `  A ) )
1614, 15ffvelrnd 5841 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( f `  x
)  e.  ( Base `  C ) )
17 simpr 458 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
y  e.  ( Base `  A ) )
1814, 17ffvelrnd 5841 . . . . . . . . 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 14632 . . . . . . . 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 2452 . . . . . . 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 2931 . . . . . 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 2931 . . . . 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 636 . . . 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 14806 . . . . . 6  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
3130breqd 4300 . . . . 5  |-  ( ph  ->  ( f ( A 
Func  C ) g  <->  f ( B  Func  D ) g ) )
3231anbi1d 699 . . . 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 14816 . . 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 2441 . . . 4  |-  ( Base `  B )  =  (
Base `  B )
3635, 9isfull 14816 . . 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 4934 1  |-  ( ph  ->  ( A Full  C )  =  ( B Full  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   class class class wbr 4289   ran crn 4837   ` cfv 5415  (class class class)co 6090   Basecbs 14170   Hom chom 14245   Hom f chomf 14600  compfccomf 14601    Func cfunc 14760   Full cful 14808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-map 7212  df-ixp 7260  df-cat 14602  df-cid 14603  df-homf 14604  df-comf 14605  df-func 14764  df-full 14810
This theorem is referenced by: (None)
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