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Theorem rescabs 13710
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 2283 . . . 4  |-  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat  J )
2 ovex 5883 . . . . 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 4160 . . . . 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 13705 . . 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 2283 . . . . . . . 8  |-  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )
14 baseid 13190 . . . . . . . . 9  |-  Base  = Slot  ( Base `  ndx )
15 1re 8837 . . . . . . . . . . 11  |-  1  e.  RR
16 1nn 9757 . . . . . . . . . . . 12  |-  1  e.  NN
17 4nn0 9984 . . . . . . . . . . . 12  |-  4  e.  NN0
18 1nn0 9981 . . . . . . . . . . . 12  |-  1  e.  NN0
19 1lt10 9930 . . . . . . . . . . . 12  |-  1  <  10
2016, 17, 18, 19declti 10149 . . . . . . . . . . 11  |-  1  < ; 1
4
2115, 20ltneii 8931 . . . . . . . . . 10  |-  1  =/= ; 1 4
22 basendx 13193 . . . . . . . . . . 11  |-  ( Base `  ndx )  =  1
23 df-hom 13232 . . . . . . . . . . . 12  |-  Hom  = Slot ; 1 4
24 4nn 9879 . . . . . . . . . . . . 13  |-  4  e.  NN
2518, 24decnncl 10137 . . . . . . . . . . . 12  |- ; 1 4  e.  NN
2623, 25ndxarg 13168 . . . . . . . . . . 11  |-  (  Hom  `  ndx )  = ; 1 4
2722, 26neeq12i 2458 . . . . . . . . . 10  |-  ( (
Base `  ndx )  =/=  (  Hom  `  ndx ) 
<->  1  =/= ; 1 4 )
2821, 27mpbir 200 . . . . . . . . 9  |-  ( Base `  ndx )  =/=  (  Hom  `  ndx )
2914, 28setsnid 13188 . . . . . . . 8  |-  ( Base `  ( Cs  S ) )  =  ( Base `  (
( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
3013, 29ressid2 13196 . . . . . . 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 5873 . . . . 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 5883 . . . . . 6  |-  ( Cs  S )  e.  _V
34 xpexg 4800 . . . . . . . . 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 5741 . . . . . . . 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 13175 . . . . . 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 2283 . . . . . . . . . . . . . 14  |-  ( Cs  S )  =  ( Cs  S )
42 eqid 2283 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
4341, 42ressbas 13198 . . . . . . . . . . . . 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 3205 . . . . . . . . . . 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 3390 . . . . . . . . . . 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 3393 . . . . . . . . 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 3394 . . . . . . . . . 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 3196 . . . . . . . 8  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( S  i^i  ( Base `  C
) )  =  ( T  i^i  ( Base `  C ) ) )
5453oveq2d 5874 . . . . . . 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 13204 . . . . . . . 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 13204 . . . . . . . 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 2325 . . . . . 6  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  ( Cs  S )  =  ( Cs  T ) )
6160oveq1d 5873 . . . . 5  |-  ( (
ph  /\  ( Base `  ( Cs  S ) )  C_  T )  ->  (
( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  J >. )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
6232, 40, 613eqtrd 2319 . . . 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 13197 . . . . . . . 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 2528 . . . . . . . . 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 4800 . . . . . . . . . . 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 5741 . . . . . . . . . 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 5539 . . . . . . . . . 10  |-  ( Base `  ( Cs  S ) )  e. 
_V
7877inex2 4156 . . . . . . . . 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 5539 . . . . . . . . 9  |-  (  Hom  `  ndx )  e.  _V
81 fvex 5539 . . . . . . . . 9  |-  ( Base `  ndx )  e.  _V
8280, 81setscom 13176 . . . . . . . 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 2283 . . . . . . . . . . 11  |-  ( ( Cs  S )s  T )  =  ( ( Cs  S )s  T )
85 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  ( Cs  S ) )  =  ( Base `  ( Cs  S ) )
8684, 85ressval2 13197 . . . . . . . . . 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 13206 . . . . . . . . . 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 2317 . . . . . . . 8  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( Cs  S ) sSet  <. ( Base `  ndx ) ,  ( T  i^i  ( Base `  ( Cs  S ) ) )
>. )  =  ( Cs  T ) )
9392oveq1d 5873 . . . . . . 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 2319 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )s  T )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
9594oveq1d 5873 . . . . 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 5883 . . . . . 6  |-  ( Cs  T )  e.  _V
9737adantr 451 . . . . . 6  |-  ( (
ph  /\  -.  ( Base `  ( Cs  S ) )  C_  T )  ->  J  e.  _V )
98 setsabs 13175 . . . . . 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 2315 . . . 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 2315 . 2  |-  ( ph  ->  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat 
J )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
103 eqid 2283 . . . 4  |-  ( C  |`cat 
H )  =  ( C  |`cat  H )
104 rescabs.c . . . 4  |-  ( ph  ->  C  e.  V )
105103, 104, 5, 71rescval2 13705 . . 3  |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. ) )
106105oveq1d 5873 . 2  |-  ( ph  ->  ( ( C  |`cat  H
)  |`cat  J )  =  ( ( ( Cs  S ) sSet  <. (  Hom  `  ndx ) ,  H >. )  |`cat 
J ) )
107 eqid 2283 . . 3  |-  ( C  |`cat 
J )  =  ( C  |`cat  J )
108107, 104, 7, 8rescval2 13705 . 2  |-  ( ph  ->  ( C  |`cat  J )  =  ( ( Cs  T ) sSet  <. (  Hom  `  ndx ) ,  J >. ) )
109102, 106, 1083eqtr4d 2325 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    i^i cin 3151    C_ wss 3152   <.cop 3643    X. cxp 4687    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   1c1 8738   4c4 9797  ;cdc 10124   ndxcnx 13145   sSet csts 13146   Basecbs 13148   ↾s cress 13149    Hom chom 13219    |`cat cresc 13685
This theorem is referenced by:  subsubc  13727
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-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-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-hom 13232  df-resc 13688
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