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Theorem resscatc 13937
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 2283 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 resscatc.2 . . . . . . . 8  |-  ( ph  ->  V  C_  U )
4 resscatc.1 . . . . . . . 8  |-  ( ph  ->  U  e.  W )
5 ssexg 4160 . . . . . . . 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 2283 . . . . . 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 13929 . . . . . . . 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 2360 . . . . . 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 2360 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( Base `  D
) )
151, 2, 7, 8, 12, 14catchom 13931 . . . . 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 2283 . . . . . 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 2283 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
2016, 17, 4catcbas 13929 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  C
)  =  ( U  i^i  Cat ) )
2120ineq2d 3370 . . . . . . . . . . 11  |-  ( ph  ->  ( V  i^i  ( Base `  C ) )  =  ( V  i^i  ( U  i^i  Cat )
) )
22 inass 3379 . . . . . . . . . . 11  |-  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  ( U  i^i  Cat ) )
2321, 22syl6reqr 2334 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  ( Base `  C )
) )
24 df-ss 3166 . . . . . . . . . . . 12  |-  ( V 
C_  U  <->  ( V  i^i  U )  =  V )
253, 24sylib 188 . . . . . . . . . . 11  |-  ( ph  ->  ( V  i^i  U
)  =  V )
2625ineq1d 3369 . . . . . . . . . 10  |-  ( ph  ->  ( ( V  i^i  U )  i^i  Cat )  =  ( V  i^i  Cat ) )
27 eqid 2283 . . . . . . . . . . . 12  |-  ( Cs  V )  =  ( Cs  V )
2827, 17ressbas 13198 . . . . . . . . . . 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 2323 . . . . . . . . 9  |-  ( ph  ->  ( V  i^i  Cat )  =  ( Base `  ( Cs  V ) ) )
3127, 17ressbasss 13200 . . . . . . . . 9  |-  ( Base `  ( Cs  V ) )  C_  ( Base `  C )
3230, 31syl6eqss 3228 . . . . . . . 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 3181 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  x  e.  ( Base `  C
) )
3533, 13sseldd 3181 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  y  e.  ( Base `  C
) )
3616, 17, 18, 19, 34, 35catchom 13931 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x (  Hom  `  C
) y )  =  ( x  Func  y
) )
3727, 19resshom 13323 . . . . . . 7  |-  ( V  e.  _V  ->  (  Hom  `  C )  =  (  Hom  `  ( Cs  V ) ) )
386, 37syl 15 . . . . . 6  |-  ( ph  ->  (  Hom  `  C
)  =  (  Hom  `  ( Cs  V ) ) )
3938proplem3 13593 . . . . 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 2322 . . . 4  |-  ( (
ph  /\  ( x  e.  ( V  i^i  Cat )  /\  y  e.  ( V  i^i  Cat )
) )  ->  (
x (  Hom  `  ( Cs  V ) ) y )  =  ( x (  Hom  `  D
) y ) )
4140ralrimivva 2635 . . 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 2283 . . . 4  |-  (  Hom  `  ( Cs  V ) )  =  (  Hom  `  ( Cs  V ) )
4310eqcomd 2288 . . . 4  |-  ( ph  ->  ( V  i^i  Cat )  =  ( Base `  D ) )
4442, 8, 30, 43homfeq 13597 . . 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 2283 . . . . . . . 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 2360 . . . . . . . 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 2360 . . . . . . . 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 2360 . . . . . . . 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 13931 . . . . . . . . 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 2359 . . . . . . . 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 13931 . . . . . . . . 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 2359 . . . . . . . 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 13933 . . . . . . 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 2283 . . . . . . . 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 3181 . . . . . . . 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 3181 . . . . . . . 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 3181 . . . . . . . 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 13933 . . . . . . 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 13324 . . . . . . . . . . 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 5875 . . . . . . . 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 5875 . . . . . . 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 2321 . . . . . 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 2635 . . . . 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 2636 . . . 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 2283 . . . . 5  |-  (comp `  ( Cs  V ) )  =  (comp `  ( Cs  V
) )
7845eqcomd 2288 . . . . 5  |-  ( ph  ->  (  Homf 
`  D )  =  (  Homf 
`  ( Cs  V ) ) )
7947, 77, 8, 43, 30, 78comfeq 13609 . . . 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 2288 . 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   <.cop 3643   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149    Hom chom 13219  compcco 13220   Catccat 13566    Homf chomf 13568  compfccomf 13569    Func cfunc 13728    o.func ccofu 13730  CatCatccatc 13926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-hom 13232  df-cco 13233  df-homf 13572  df-comf 13573  df-catc 13927
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