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Theorem fullpropd 13810
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 13798 . . 3  |-  Rel  ( A Full  C )
21a1i 10 . 2  |-  ( ph  ->  Rel  ( A Full  C
) )
3 relfull 13798 . . 3  |-  Rel  ( B Full  D )
43a1i 10 . 2  |-  ( ph  ->  Rel  ( B Full  D
) )
5 fullpropd.1 . . . . . . . 8  |-  ( ph  ->  (  Homf 
`  A )  =  (  Homf 
`  B ) )
65homfeqbas 13615 . . . . . . 7  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
76adantr 451 . . . . . 6  |-  ( (
ph  /\  f ( A  Func  C ) g )  ->  ( Base `  A )  =  (
Base `  B )
)
87adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  f
( A  Func  C
) g )  /\  x  e.  ( Base `  A ) )  -> 
( Base `  A )  =  ( Base `  B
) )
9 eqid 2296 . . . . . . . . 9  |-  ( Base `  C )  =  (
Base `  C )
10 eqid 2296 . . . . . . . . 9  |-  (  Hom  `  C )  =  (  Hom  `  C )
11 eqid 2296 . . . . . . . . 9  |-  (  Hom  `  D )  =  (  Hom  `  D )
12 fullpropd.3 . . . . . . . . . 10  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
1312ad3antrrr 710 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
(  Homf 
`  C )  =  (  Homf 
`  D ) )
14 eqid 2296 . . . . . . . . . . 11  |-  ( Base `  A )  =  (
Base `  A )
15 simpllr 735 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
f ( A  Func  C ) g )
1614, 9, 15funcf1 13756 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
f : ( Base `  A ) --> ( Base `  C ) )
17 simplr 731 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  ->  x  e.  ( Base `  A ) )
1816, 17ffvelrnd 5682 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( f `  x
)  e.  ( Base `  C ) )
19 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
y  e.  ( Base `  A ) )
2016, 19ffvelrnd 5682 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f ( A  Func  C ) g )  /\  x  e.  ( Base `  A ) )  /\  y  e.  ( Base `  A ) )  -> 
( f `  y
)  e.  ( Base `  C ) )
219, 10, 11, 13, 18, 20homfeqval 13616 . . . . . . . 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
) ) )
2221eqeq2d 2307 . . . . . . 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 ) ) ) )
238, 22raleqbidva 2763 . . . . . 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 ) ) ) )
247, 23raleqbidva 2763 . . . . 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
) ) ) )
2524pm5.32da 622 . . . 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 ) ) ) ) )
26 fullpropd.2 . . . . . . 7  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
27 fullpropd.4 . . . . . . 7  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
28 fullpropd.a . . . . . . 7  |-  ( ph  ->  A  e.  V )
29 fullpropd.b . . . . . . 7  |-  ( ph  ->  B  e.  V )
30 fullpropd.c . . . . . . 7  |-  ( ph  ->  C  e.  V )
31 fullpropd.d . . . . . . 7  |-  ( ph  ->  D  e.  V )
325, 26, 12, 27, 28, 29, 30, 31funcpropd 13790 . . . . . 6  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
3332breqd 4050 . . . . 5  |-  ( ph  ->  ( f ( A 
Func  C ) g  <->  f ( B  Func  D ) g ) )
3433anbi1d 685 . . . 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 ) ) ) ) )
3525, 34bitrd 244 . . 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 ) ) ) ) )
3614, 10isfull 13800 . . 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 ) ) ) )
37 eqid 2296 . . . 4  |-  ( Base `  B )  =  (
Base `  B )
3837, 11isfull 13800 . . 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 ) ) ) )
3935, 36, 383bitr4g 279 . 2  |-  ( ph  ->  ( f ( A Full 
C ) g  <->  f ( B Full  D ) g ) )
402, 4, 39eqbrrdv 4800 1  |-  ( ph  ->  ( A Full  C )  =  ( B Full  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   ran crn 4706   Rel wrel 4710   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235    Homf chomf 13584  compfccomf 13585    Func cfunc 13744   Full cful 13792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-map 6790  df-ixp 6834  df-cat 13586  df-cid 13587  df-homf 13588  df-comf 13589  df-func 13748  df-full 13794
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