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Theorem resscatc 13953
Description: The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCat `  U categories for different  U are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resscatc.c  |-  C  =  (CatCat `  U )
resscatc.d  |-  D  =  (CatCat `  V )
resscatc.1  |-  ( ph  ->  U  e.  W )
resscatc.2  |-  ( ph  ->  V  C_  U )
Assertion
Ref Expression
resscatc  |-  ( ph  ->  ( (  Homf  `  ( Cs  V ) )  =  (  Homf 
`  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )

Proof of Theorem resscatc
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resscatc.d . . . . . 6  |-  D  =  (CatCat `  V )
2 eqid 2296 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 resscatc.2 . . . . . . . 8  |-  ( ph  ->  V  C_  U )
4 resscatc.1 . . . . . . . 8  |-  ( ph  ->  U  e.  W )
5 ssexg 4176 . . . . . . . 8  |-  ( ( V  C_  U  /\  U  e.  W )  ->  V  e.  _V )
63, 4, 5syl2anc 642 . . . . . . 7  |-  ( ph  ->  V  e.  _V )
76adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  V  e.  _V )
8 eqid 2296 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
9 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( V  i^i  Cat ) )
101, 2, 6catcbas 13945 . . . . . . . 8  |-  ( ph  ->  ( Base `  D
)  =  ( V  i^i  Cat ) )
1110adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  ( Base `  D )  =  ( V  i^i  Cat ) )
129, 11eleqtrrd 2373 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( Base `  D
) )
13 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( V  i^i  Cat ) )
1413, 11eleqtrrd 2373 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( Base `  D
) )
151, 2, 7, 8, 12, 14catchom 13947 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x (  Hom  `  D
) y )  =  ( x  Func  y
) )
16 resscatc.c . . . . . 6  |-  C  =  (CatCat `  U )
17 eqid 2296 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
184adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  U  e.  W )
19 eqid 2296 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
2016, 17, 4catcbas 13945 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  C
)  =  ( U  i^i  Cat ) )
2120ineq2d 3383 . . . . . . . . . . 11  |-  ( ph  ->  ( V  i^i  ( Base `  C ) )  =  ( V  i^i  ( U  i^i  Cat )
) )
22 inass 3392 . . . . . . . . . . 11  |-  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  ( U  i^i  Cat ) )
2321, 22syl6reqr 2347 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  ( Base `  C )
) )
24 df-ss 3179 . . . . . . . . . . . 12  |-  ( V 
C_  U  <->  ( V  i^i  U )  =  V )
253, 24sylib 188 . . . . . . . . . . 11  |-  ( ph  ->  ( V  i^i  U
)  =  V )
2625ineq1d 3382 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  Cat ) )
27 eqid 2296 . . . . . . . . . . . 12  |-  ( Cs  V )  =  ( Cs  V )
2827, 17ressbas 13214 . . . . . . . . . . 11  |-  ( V  e.  _V  ->  ( V  i^i  ( Base `  C
) )  =  (
Base `  ( Cs  V
) ) )
296, 28syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( V  i^i  ( Base `  C ) )  =  ( Base `  ( Cs  V ) ) )
3023, 26, 293eqtr3d 2336 . . . . . . . . 9  |-  ( ph  ->  ( V  i^i  Cat )  =  ( Base `  ( Cs  V ) ) )
3127, 17ressbasss 13216 . . . . . . . . 9  |-  ( Base `  ( Cs  V ) )  C_  ( Base `  C )
3230, 31syl6eqss 3241 . . . . . . . 8  |-  ( ph  ->  ( V  i^i  Cat )  C_  ( Base `  C
) )
3332adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  ( V  i^i  Cat )  C_  ( Base `  C )
)
3433, 9sseldd 3194 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( Base `  C
) )
3533, 13sseldd 3194 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( Base `  C
) )
3616, 17, 18, 19, 34, 35catchom 13947 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x (  Hom  `  C
) y )  =  ( x  Func  y
) )
3727, 19resshom 13339 . . . . . . 7  |-  ( V  e.  _V  ->  (  Hom  `  C )  =  (  Hom  `  ( Cs  V ) ) )
386, 37syl 15 . . . . . 6  |-  ( ph  ->  (  Hom  `  C
)  =  (  Hom  `  ( Cs  V ) ) )
3938proplem3 13609 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x (  Hom  `  C
) y )  =  ( x (  Hom  `  ( Cs  V ) ) y ) )
4015, 36, 393eqtr2rd 2335 . . . 4  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x (  Hom  `  ( Cs  V ) ) y )  =  ( x (  Hom  `  D
) y ) )
4140ralrimivva 2648 . . 3  |-  ( ph  ->  A. x  e.  ( V  i^i  Cat ) A. y  e.  ( V  i^i  Cat ) ( x (  Hom  `  ( Cs  V ) ) y )  =  ( x (  Hom  `  D
) y ) )
42 eqid 2296 . . . 4  |-  (  Hom  `  ( Cs  V ) )  =  (  Hom  `  ( Cs  V ) )
4310eqcomd 2301 . . . 4  |-  ( ph  ->  ( V  i^i  Cat )  =  ( Base `  D ) )
4442, 8, 30, 43homfeq 13613 . . 3  |-  ( ph  ->  ( (  Homf  `  ( Cs  V ) )  =  (  Homf 
`  D )  <->  A. x  e.  ( V  i^i  Cat ) A. y  e.  ( V  i^i  Cat )
( x (  Hom  `  ( Cs  V ) ) y )  =  ( x (  Hom  `  D
) y ) ) )
4541, 44mpbird 223 . 2  |-  ( ph  ->  (  Homf 
`  ( Cs  V ) )  =  (  Homf  `  D ) )
466ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  V  e.  _V )
47 eqid 2296 . . . . . . . 8  |-  (comp `  D )  =  (comp `  D )
48 simplr1 997 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  x  e.  ( V  i^i  Cat )
)
4910ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  ( Base `  D
)  =  ( V  i^i  Cat ) )
5048, 49eleqtrrd 2373 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  x  e.  (
Base `  D )
)
51 simplr2 998 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  y  e.  ( V  i^i  Cat )
)
5251, 49eleqtrrd 2373 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  y  e.  (
Base `  D )
)
53 simplr3 999 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  z  e.  ( V  i^i  Cat )
)
5453, 49eleqtrrd 2373 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  z  e.  (
Base `  D )
)
55 simprl 732 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  f  e.  ( x (  Hom  `  D
) y ) )
561, 2, 46, 8, 50, 52catchom 13947 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  ( x (  Hom  `  D )
y )  =  ( x  Func  y )
)
5755, 56eleqtrd 2372 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  f  e.  ( x  Func  y )
)
58 simprr 733 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  g  e.  ( y (  Hom  `  D
) z ) )
591, 2, 46, 8, 52, 54catchom 13947 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  ( y (  Hom  `  D )
z )  =  ( y  Func  z )
)
6058, 59eleqtrd 2372 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  g  e.  ( y  Func  z )
)
611, 2, 46, 47, 50, 52, 54, 57, 60catcco 13949 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  ( g (
<. x ,  y >.
(comp `  D )
z ) f )  =  ( g  o.func  f ) )
624ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  U  e.  W
)
63 eqid 2296 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
6432ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  ( V  i^i  Cat )  C_  ( Base `  C ) )
6564, 48sseldd 3194 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  x  e.  (
Base `  C )
)
6664, 51sseldd 3194 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  y  e.  (
Base `  C )
)
6764, 53sseldd 3194 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  z  e.  (
Base `  C )
)
6816, 17, 62, 63, 65, 66, 67, 57, 60catcco 13949 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  ( g (
<. x ,  y >.
(comp `  C )
z ) f )  =  ( g  o.func  f ) )
6927, 63ressco 13340 . . . . . . . . . . 11  |-  ( V  e.  _V  ->  (comp `  C )  =  (comp `  ( Cs  V ) ) )
706, 69syl 15 . . . . . . . . . 10  |-  ( ph  ->  (comp `  C )  =  (comp `  ( Cs  V
) ) )
7170ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  (comp `  C
)  =  (comp `  ( Cs  V ) ) )
7271oveqd 5891 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  ( <. x ,  y >. (comp `  C ) z )  =  ( <. x ,  y >. (comp `  ( Cs  V ) ) z ) )
7372oveqd 5891 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  ( g (
<. x ,  y >.
(comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  ( Cs  V
) ) z ) f ) )
7461, 68, 733eqtr2d 2334 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  /\  ( f  e.  ( x (  Hom  `  D
) y )  /\  g  e.  ( y
(  Hom  `  D ) z ) ) )  ->  ( g (
<. x ,  y >.
(comp `  D )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  ( Cs  V
) ) z ) f ) )
7574ralrimivva 2648 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )  /\  z  e.  ( V  i^i  Cat ) ) )  ->  A. f  e.  ( x (  Hom  `  D ) y ) A. g  e.  ( y (  Hom  `  D
) z ) ( g ( <. x ,  y >. (comp `  D ) z ) f )  =  ( g ( <. x ,  y >. (comp `  ( Cs  V ) ) z ) f ) )
7675ralrimivvva 2649 . . . 4  |-  ( ph  ->  A. x  e.  ( V  i^i  Cat ) A. y  e.  ( V  i^i  Cat ) A. z  e.  ( V  i^i  Cat ) A. f  e.  ( x (  Hom  `  D ) y ) A. g  e.  ( y (  Hom  `  D
) z ) ( g ( <. x ,  y >. (comp `  D ) z ) f )  =  ( g ( <. x ,  y >. (comp `  ( Cs  V ) ) z ) f ) )
77 eqid 2296 . . . . 5  |-  (comp `  ( Cs  V ) )  =  (comp `  ( Cs  V
) )
7845eqcomd 2301 . . . . 5  |-  ( ph  ->  (  Homf 
`  D )  =  (  Homf 
`  ( Cs  V ) ) )
7947, 77, 8, 43, 30, 78comfeq 13625 . . . 4  |-  ( ph  ->  ( (compf `  D )  =  (compf `  ( Cs  V ) )  <->  A. x  e.  ( V  i^i  Cat ) A. y  e.  ( V  i^i  Cat ) A. z  e.  ( V  i^i  Cat ) A. f  e.  ( x
(  Hom  `  D ) y ) A. g  e.  ( y (  Hom  `  D ) z ) ( g ( <.
x ,  y >.
(comp `  D )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  ( Cs  V
) ) z ) f ) ) )
8076, 79mpbird 223 . . 3  |-  ( ph  ->  (compf `  D )  =  (compf `  ( Cs  V ) ) )
8180eqcomd 2301 . 2  |-  ( ph  ->  (compf `  ( Cs  V ) )  =  (compf `  D ) )
8245, 81jca 518 1  |-  ( ph  ->  ( (  Homf  `  ( Cs  V ) )  =  (  Homf 
`  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165   <.cop 3656   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165    Hom chom 13235  compcco 13236   Catccat 13582    Homf chomf 13584  compfccomf 13585    Func cfunc 13744    o.func ccofu 13746  CatCatccatc 13942
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  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-hom 13248  df-cco 13249  df-homf 13588  df-comf 13589  df-catc 13943
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