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Theorem rescabs 13726
Description: Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
rescabs.c  |-  ( ph  ->  C  e.  V )
rescabs.h  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
rescabs.j  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
rescabs.s  |-  ( ph  ->  S  e.  W )
rescabs.t  |-  ( ph  ->  T  C_  S )
Assertion
Ref Expression
rescabs  |-  ( ph  ->  ( ( C  |`cat  H
)  |`cat  J )  =  ( C  |`cat  J ) )

Proof of Theorem rescabs
StepHypRef Expression
1 eqid 2296 . . . 4  |-  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat  J )
2 ovex 5899 . . . . 5  |-  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V
32a1i 10 . . . 4  |-  ( ph  ->  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V )
4 rescabs.t . . . . 5  |-  ( ph  ->  T  C_  S )
5 rescabs.s . . . . 5  |-  ( ph  ->  S  e.  W )
6 ssexg 4176 . . . . 5  |-  ( ( T  C_  S  /\  S  e.  W )  ->  T  e.  _V )
74, 5, 6syl2anc 642 . . . 4  |-  ( ph  ->  T  e.  _V )
8 rescabs.j . . . 4  |-  ( ph  ->  J  Fn  ( T  X.  T ) )
91, 3, 7, 8rescval2 13721 . . 3  |-  ( ph  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat 
J )  =  ( ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
10 simpr 447 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Base `  ( Cs  S ) )  C_  T )
112a1i 10 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V )
127adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  T  e.  _V )
13 eqid 2296 . . . . . . . 8  |-  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )
14 baseid 13206 . . . . . . . . 9  |-  Base  = Slot  ( Base `  ndx )
15 1re 8853 . . . . . . . . . . 11  |-  1  e.  RR
16 1nn 9773 . . . . . . . . . . . 12  |-  1  e.  NN
17 4nn0 10000 . . . . . . . . . . . 12  |-  4  e.  NN0
18 1nn0 9997 . . . . . . . . . . . 12  |-  1  e.  NN0
19 1lt10 9946 . . . . . . . . . . . 12  |-  1  <  10
2016, 17, 18, 19declti 10165 . . . . . . . . . . 11  |-  1  < ; 1
4
2115, 20ltneii 8947 . . . . . . . . . 10  |-  1  =/= ; 1 4
22 basendx 13209 . . . . . . . . . . 11  |-  ( Base `  ndx )  =  1
23 df-hom 13248 . . . . . . . . . . . 12  |-  Hom  = Slot ; 1 4
24 4nn 9895 . . . . . . . . . . . . 13  |-  4  e.  NN
2518, 24decnncl 10153 . . . . . . . . . . . 12  |- ; 1 4  e.  NN
2623, 25ndxarg 13184 . . . . . . . . . . 11  |-  (  Hom  `  ndx )  = ; 1 4
2722, 26neeq12i 2471 . . . . . . . . . 10  |-  ( (
Base `  ndx )  =/=  (  Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
2821, 27mpbir 200 . . . . . . . . 9  |-  ( Base `  ndx )  =/=  (  Hom  `  ndx )
2914, 28setsnid 13204 . . . . . . . 8  |-  ( Base `  ( Cs  S ) )  =  ( Base `  (
( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
3013, 29ressid2 13212 . . . . . . 7  |-  ( ( ( Base `  ( Cs  S ) )  C_  T  /\  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V  /\  T  e.  _V )  ->  (
( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
3110, 11, 12, 30syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
3231oveq1d 5889 . . . . 5  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
33 ovex 5899 . . . . . 6  |-  ( Cs  S )  e.  _V
34 xpexg 4816 . . . . . . . . 9  |-  ( ( T  e.  _V  /\  T  e.  _V )  ->  ( T  X.  T
)  e.  _V )
357, 7, 34syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( T  X.  T
)  e.  _V )
36 fnex 5757 . . . . . . . 8  |-  ( ( J  Fn  ( T  X.  T )  /\  ( T  X.  T
)  e.  _V )  ->  J  e.  _V )
378, 35, 36syl2anc 642 . . . . . . 7  |-  ( ph  ->  J  e.  _V )
3837adantr 451 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  J  e.  _V )
39 setsabs 13191 . . . . . 6  |-  ( ( ( Cs  S )  e.  _V  /\  J  e.  _V )  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
4033, 38, 39sylancr 644 . . . . 5  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
41 eqid 2296 . . . . . . . . . . . . . 14  |-  ( Cs  S )  =  ( Cs  S )
42 eqid 2296 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
4341, 42ressbas 13214 . . . . . . . . . . . . 13  |-  ( S  e.  W  ->  ( S  i^i  ( Base `  C
) )  =  (
Base `  ( Cs  S
) ) )
445, 43syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( S  i^i  ( Base `  C ) )  =  ( Base `  ( Cs  S ) ) )
4544sseq1d 3218 . . . . . . . . . . 11  |-  ( ph  ->  ( ( S  i^i  ( Base `  C )
)  C_  T  <->  ( Base `  ( Cs  S ) )  C_  T ) )
4645biimpar 471 . . . . . . . . . 10  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  C_  T
)
47 inss2 3403 . . . . . . . . . . 11  |-  ( S  i^i  ( Base `  C
) )  C_  ( Base `  C )
4847a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  C_  ( Base `  C ) )
4946, 48ssind 3406 . . . . . . . . 9  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  C_  ( T  i^i  ( Base `  C
) ) )
504adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  T  C_  S )
51 ssrin 3407 . . . . . . . . . 10  |-  ( T 
C_  S  ->  ( T  i^i  ( Base `  C
) )  C_  ( S  i^i  ( Base `  C
) ) )
5250, 51syl 15 . . . . . . . . 9  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( T  i^i  ( Base `  C
) )  C_  ( S  i^i  ( Base `  C
) ) )
5349, 52eqssd 3209 . . . . . . . 8  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  =  ( T  i^i  ( Base `  C ) ) )
5453oveq2d 5890 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  ( S  i^i  ( Base `  C ) ) )  =  ( Cs  ( T  i^i  ( Base `  C ) ) ) )
555adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  S  e.  W )
5642ressinbas 13220 . . . . . . . 8  |-  ( S  e.  W  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
5755, 56syl 15 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
5842ressinbas 13220 . . . . . . . 8  |-  ( T  e.  _V  ->  ( Cs  T )  =  ( Cs  ( T  i^i  ( Base `  C ) ) ) )
5912, 58syl 15 . . . . . . 7  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  T )  =  ( Cs  ( T  i^i  ( Base `  C ) ) ) )
6054, 57, 593eqtr4d 2338 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  S )  =  ( Cs  T ) )
6160oveq1d 5889 . . . . 5  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
6232, 40, 613eqtrd 2332 . . . 4  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
63 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  -.  ( Base `  ( Cs  S ) )  C_  T )
642a1i 10 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V )
657adantr 451 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  T  e.  _V )
6613, 29ressval2 13213 . . . . . . . 8  |-  ( ( -.  ( Base `  ( Cs  S ) )  C_  T  /\  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  e.  _V  /\  T  e.  _V )  ->  (
( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. )
)
6763, 64, 65, 66syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. )
)
6833a1i 10 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  S )  e.  _V )
6928necomi 2541 . . . . . . . . 9  |-  (  Hom  `  ndx )  =/=  ( Base `  ndx )
7069a1i 10 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  (  Hom  `  ndx )  =/=  ( Base `  ndx ) )
71 rescabs.h . . . . . . . . . 10  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
72 xpexg 4816 . . . . . . . . . . 11  |-  ( ( S  e.  W  /\  S  e.  W )  ->  ( S  X.  S
)  e.  _V )
735, 5, 72syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( S  X.  S
)  e.  _V )
74 fnex 5757 . . . . . . . . . 10  |-  ( ( H  Fn  ( S  X.  S )  /\  ( S  X.  S
)  e.  _V )  ->  H  e.  _V )
7571, 73, 74syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  H  e.  _V )
7675adantr 451 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  H  e.  _V )
77 fvex 5555 . . . . . . . . . 10  |-  ( Base `  ( Cs  S ) )  e. 
_V
7877inex2 4172 . . . . . . . . 9  |-  ( T  i^i  ( Base `  ( Cs  S ) ) )  e.  _V
7978a1i 10 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( T  i^i  ( Base `  ( Cs  S ) ) )  e.  _V )
80 fvex 5555 . . . . . . . . 9  |-  (  Hom  `  ndx )  e.  _V
81 fvex 5555 . . . . . . . . 9  |-  ( Base `  ndx )  e.  _V
8280, 81setscom 13192 . . . . . . . 8  |-  ( ( ( ( Cs  S )  e.  _V  /\  (  Hom  `  ndx )  =/=  ( Base `  ndx ) )  /\  ( H  e.  _V  /\  ( T  i^i  ( Base `  ( Cs  S ) ) )  e.  _V ) )  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. )  =  ( ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. ) sSet  <.
(  Hom  `  ndx ) ,  H >. ) )
8368, 70, 76, 79, 82syl22anc 1183 . . . . . . 7  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) )
>. )  =  (
( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) )
>. ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
84 eqid 2296 . . . . . . . . . . 11  |-  ( ( Cs  S )s  T )  =  ( ( Cs  S )s  T )
85 eqid 2296 . . . . . . . . . . 11  |-  ( Base `  ( Cs  S ) )  =  ( Base `  ( Cs  S ) )
8684, 85ressval2 13213 . . . . . . . . . 10  |-  ( ( -.  ( Base `  ( Cs  S ) )  C_  T  /\  ( Cs  S )  e.  _V  /\  T  e.  _V )  ->  (
( Cs  S )s  T )  =  ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. )
)
8763, 68, 65, 86syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S )s  T )  =  ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) ) >. )
)
885adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  S  e.  W )
894adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  T  C_  S )
90 ressabs 13222 . . . . . . . . . 10  |-  ( ( S  e.  W  /\  T  C_  S )  -> 
( ( Cs  S )s  T )  =  ( Cs  T ) )
9188, 89, 90syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S )s  T )  =  ( Cs  T ) )
9287, 91eqtr3d 2330 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) )
>. )  =  ( Cs  T ) )
9392oveq1d 5889 . . . . . . 7  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) )
>. ) sSet  <. (  Hom  `  ndx ) ,  H >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
9467, 83, 933eqtrd 2332 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
9594oveq1d 5889 . . . . 5  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
96 ovex 5899 . . . . . 6  |-  ( Cs  T )  e.  _V
9737adantr 451 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  J  e.  _V )
98 setsabs 13191 . . . . . 6  |-  ( ( ( Cs  T )  e.  _V  /\  J  e.  _V )  ->  ( ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
9996, 97, 98sylancr 644 . . . . 5  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  H >. ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
10095, 99eqtrd 2328 . . . 4  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
10162, 100pm2.61dan 766 . . 3  |-  ( ph  ->  ( ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
1029, 101eqtrd 2328 . 2  |-  ( ph  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat 
J )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
103 eqid 2296 . . . 4  |-  ( C  |`cat 
H )  =  ( C  |`cat  H )
104 rescabs.c . . . 4  |-  ( ph  ->  C  e.  V )
105103, 104, 5, 71rescval2 13721 . . 3  |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
106105oveq1d 5889 . 2  |-  ( ph  ->  ( ( C  |`cat  H
)  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat 
J ) )
107 eqid 2296 . . 3  |-  ( C  |`cat 
J )  =  ( C  |`cat  J )
108107, 104, 7, 8rescval2 13721 . 2  |-  ( ph  ->  ( C  |`cat  J )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
109102, 106, 1083eqtr4d 2338 1  |-  ( ph  ->  ( ( C  |`cat  H
)  |`cat  J )  =  ( C  |`cat  J ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    i^i cin 3164    C_ wss 3165   <.cop 3656    X. cxp 4703    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   1c1 8754   4c4 9813  ;cdc 10140   ndxcnx 13161   sSet csts 13162   Basecbs 13164   ↾s cress 13165    Hom chom 13235    |`cat cresc 13701
This theorem is referenced by:  subsubc  13743
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-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-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-hom 13248  df-resc 13704
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