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Theorem ressress 13205
Description: Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)
Assertion
Ref Expression
ressress  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B ) ) )

Proof of Theorem ressress
StepHypRef Expression
1 simplr 731 . . . . . . . . 9  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  -.  ( Base `  W
)  C_  A )
2 simpr1 961 . . . . . . . . 9  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  W  e.  _V )
3 simpr2 962 . . . . . . . . 9  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  A  e.  X )
4 eqid 2283 . . . . . . . . . 10  |-  ( Ws  A )  =  ( Ws  A )
5 eqid 2283 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
64, 5ressval2 13197 . . . . . . . . 9  |-  ( ( -.  ( Base `  W
)  C_  A  /\  W  e.  _V  /\  A  e.  X )  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )
)
71, 2, 3, 6syl3anc 1182 . . . . . . . 8  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  -> 
( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )
)
8 inass 3379 . . . . . . . . . . 11  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  =  ( A  i^i  ( B  i^i  ( Base `  W
) ) )
9 in12 3380 . . . . . . . . . . 11  |-  ( A  i^i  ( B  i^i  ( Base `  W )
) )  =  ( B  i^i  ( A  i^i  ( Base `  W
) ) )
108, 9eqtri 2303 . . . . . . . . . 10  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  =  ( B  i^i  ( A  i^i  ( Base `  W
) ) )
114, 5ressbas 13198 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  ( A  i^i  ( Base `  W
) )  =  (
Base `  ( Ws  A
) ) )
123, 11syl 15 . . . . . . . . . . 11  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  -> 
( A  i^i  ( Base `  W ) )  =  ( Base `  ( Ws  A ) ) )
1312ineq2d 3370 . . . . . . . . . 10  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  -> 
( B  i^i  ( A  i^i  ( Base `  W
) ) )  =  ( B  i^i  ( Base `  ( Ws  A ) ) ) )
1410, 13syl5req 2328 . . . . . . . . 9  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  -> 
( B  i^i  ( Base `  ( Ws  A ) ) )  =  ( ( A  i^i  B
)  i^i  ( Base `  W ) ) )
1514opeq2d 3803 . . . . . . . 8  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) )
>.  =  <. ( Base `  ndx ) ,  ( ( A  i^i  B
)  i^i  ( Base `  W ) ) >.
)
167, 15oveq12d 5876 . . . . . . 7  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  -> 
( ( Ws  A ) sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) )
>. )  =  (
( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W ) ) >.
) sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W
) ) >. )
)
17 fvex 5539 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
1817inex2 4156 . . . . . . . 8  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  e.  _V
19 setsabs 13175 . . . . . . . 8  |-  ( ( W  e.  _V  /\  ( ( A  i^i  B )  i^i  ( Base `  W ) )  e. 
_V )  ->  (
( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W ) ) >.
) sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W
) ) >. )  =  ( W sSet  <. (
Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W ) ) >.
) )
202, 18, 19sylancl 643 . . . . . . 7  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  -> 
( ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  ( Base `  W ) )
>. ) sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B
)  i^i  ( Base `  W ) ) >.
)  =  ( W sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W
) ) >. )
)
2116, 20eqtrd 2315 . . . . . 6  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  -> 
( ( Ws  A ) sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) )
>. )  =  ( W sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W
) ) >. )
)
22 simpll 730 . . . . . . 7  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  -.  ( Base `  ( Ws  A ) )  C_  B )
23 ovex 5883 . . . . . . . 8  |-  ( Ws  A )  e.  _V
2423a1i 10 . . . . . . 7  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  -> 
( Ws  A )  e.  _V )
25 simpr3 963 . . . . . . 7  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  B  e.  Y )
26 eqid 2283 . . . . . . . 8  |-  ( ( Ws  A )s  B )  =  ( ( Ws  A )s  B )
27 eqid 2283 . . . . . . . 8  |-  ( Base `  ( Ws  A ) )  =  ( Base `  ( Ws  A ) )
2826, 27ressval2 13197 . . . . . . 7  |-  ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  ( Ws  A )  e.  _V  /\  B  e.  Y )  ->  (
( Ws  A )s  B )  =  ( ( Ws  A ) sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) ) >. )
)
2922, 24, 25, 28syl3anc 1182 . . . . . 6  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  -> 
( ( Ws  A )s  B )  =  ( ( Ws  A ) sSet  <. ( Base `  ndx ) ,  ( B  i^i  ( Base `  ( Ws  A ) ) ) >. )
)
30 inss1 3389 . . . . . . . . 9  |-  ( A  i^i  B )  C_  A
31 sstr 3187 . . . . . . . . 9  |-  ( ( ( Base `  W
)  C_  ( A  i^i  B )  /\  ( A  i^i  B )  C_  A )  ->  ( Base `  W )  C_  A )
3230, 31mpan2 652 . . . . . . . 8  |-  ( (
Base `  W )  C_  ( A  i^i  B
)  ->  ( Base `  W )  C_  A
)
331, 32nsyl 113 . . . . . . 7  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  -.  ( Base `  W
)  C_  ( A  i^i  B ) )
34 inex1g 4157 . . . . . . . 8  |-  ( A  e.  X  ->  ( A  i^i  B )  e. 
_V )
353, 34syl 15 . . . . . . 7  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  -> 
( A  i^i  B
)  e.  _V )
36 eqid 2283 . . . . . . . 8  |-  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( A  i^i  B ) )
3736, 5ressval2 13197 . . . . . . 7  |-  ( ( -.  ( Base `  W
)  C_  ( A  i^i  B )  /\  W  e.  _V  /\  ( A  i^i  B )  e. 
_V )  ->  ( Ws  ( A  i^i  B ) )  =  ( W sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W
) ) >. )
)
3833, 2, 35, 37syl3anc 1182 . . . . . 6  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  -> 
( Ws  ( A  i^i  B ) )  =  ( W sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  ( Base `  W
) ) >. )
)
3921, 29, 383eqtr4d 2325 . . . . 5  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  -> 
( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B ) ) )
4039exp31 587 . . . 4  |-  ( -.  ( Base `  ( Ws  A ) )  C_  B  ->  ( -.  ( Base `  W )  C_  A  ->  ( ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )  ->  (
( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) ) ) )
4126, 27ressid2 13196 . . . . . . . 8  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( Ws  A )  e.  _V  /\  B  e.  Y )  ->  (
( Ws  A )s  B )  =  ( Ws  A ) )
4223, 41mp3an2 1265 . . . . . . 7  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  B  e.  Y
)  ->  ( ( Ws  A )s  B )  =  ( Ws  A ) )
43423ad2antr3 1122 . . . . . 6  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  (
( Ws  A )s  B )  =  ( Ws  A ) )
44 in32 3381 . . . . . . . . 9  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  =  ( ( A  i^i  ( Base `  W ) )  i^i  B )
45 simpr2 962 . . . . . . . . . . . 12  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  A  e.  X )
4645, 11syl 15 . . . . . . . . . . 11  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( A  i^i  ( Base `  W
) )  =  (
Base `  ( Ws  A
) ) )
47 simpl 443 . . . . . . . . . . 11  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( Base `  ( Ws  A ) )  C_  B )
4846, 47eqsstrd 3212 . . . . . . . . . 10  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( A  i^i  ( Base `  W
) )  C_  B
)
49 df-ss 3166 . . . . . . . . . 10  |-  ( ( A  i^i  ( Base `  W ) )  C_  B 
<->  ( ( A  i^i  ( Base `  W )
)  i^i  B )  =  ( A  i^i  ( Base `  W )
) )
5048, 49sylib 188 . . . . . . . . 9  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  (
( A  i^i  ( Base `  W ) )  i^i  B )  =  ( A  i^i  ( Base `  W ) ) )
5144, 50syl5req 2328 . . . . . . . 8  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( A  i^i  ( Base `  W
) )  =  ( ( A  i^i  B
)  i^i  ( Base `  W ) ) )
5251oveq2d 5874 . . . . . . 7  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( Ws  ( A  i^i  ( Base `  W ) ) )  =  ( Ws  ( ( A  i^i  B
)  i^i  ( Base `  W ) ) ) )
535ressinbas 13204 . . . . . . . 8  |-  ( A  e.  X  ->  ( Ws  A )  =  ( Ws  ( A  i^i  ( Base `  W ) ) ) )
5445, 53syl 15 . . . . . . 7  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  ( Base `  W ) ) ) )
555ressinbas 13204 . . . . . . . 8  |-  ( ( A  i^i  B )  e.  _V  ->  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( ( A  i^i  B
)  i^i  ( Base `  W ) ) ) )
5645, 34, 553syl 18 . . . . . . 7  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( ( A  i^i  B
)  i^i  ( Base `  W ) ) ) )
5752, 54, 563eqtr4d 2325 . . . . . 6  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
5843, 57eqtrd 2315 . . . . 5  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  (
( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) )
5958ex 423 . . . 4  |-  ( (
Base `  ( Ws  A
) )  C_  B  ->  ( ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
)  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) ) )
604, 5ressid2 13196 . . . . . . . 8  |-  ( ( ( Base `  W
)  C_  A  /\  W  e.  _V  /\  A  e.  X )  ->  ( Ws  A )  =  W )
61603adant3r3 1162 . . . . . . 7  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Ws  A
)  =  W )
6261oveq1d 5873 . . . . . 6  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( ( Ws  A )s  B )  =  ( Ws  B ) )
63 inss2 3390 . . . . . . . . . . 11  |-  ( B  i^i  ( Base `  W
) )  C_  ( Base `  W )
64 simpl 443 . . . . . . . . . . 11  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Base `  W )  C_  A
)
6563, 64syl5ss 3190 . . . . . . . . . 10  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( B  i^i  ( Base `  W
) )  C_  A
)
66 sseqin2 3388 . . . . . . . . . 10  |-  ( ( B  i^i  ( Base `  W ) )  C_  A 
<->  ( A  i^i  ( B  i^i  ( Base `  W
) ) )  =  ( B  i^i  ( Base `  W ) ) )
6765, 66sylib 188 . . . . . . . . 9  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( A  i^i  ( B  i^i  ( Base `  W ) ) )  =  ( B  i^i  ( Base `  W
) ) )
688, 67syl5req 2328 . . . . . . . 8  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( B  i^i  ( Base `  W
) )  =  ( ( A  i^i  B
)  i^i  ( Base `  W ) ) )
6968oveq2d 5874 . . . . . . 7  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Ws  ( B  i^i  ( Base `  W
) ) )  =  ( Ws  ( ( A  i^i  B )  i^i  ( Base `  W
) ) ) )
70 simpr3 963 . . . . . . . 8  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  B  e.  Y )
715ressinbas 13204 . . . . . . . 8  |-  ( B  e.  Y  ->  ( Ws  B )  =  ( Ws  ( B  i^i  ( Base `  W ) ) ) )
7270, 71syl 15 . . . . . . 7  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Ws  B
)  =  ( Ws  ( B  i^i  ( Base `  W ) ) ) )
73 simpr2 962 . . . . . . . 8  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  A  e.  X )
7473, 34, 553syl 18 . . . . . . 7  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( ( A  i^i  B )  i^i  ( Base `  W
) ) ) )
7569, 72, 743eqtr4d 2325 . . . . . 6  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Ws  B
)  =  ( Ws  ( A  i^i  B ) ) )
7662, 75eqtrd 2315 . . . . 5  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) )
7776ex 423 . . . 4  |-  ( (
Base `  W )  C_  A  ->  ( ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )  ->  (
( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) ) )
7840, 59, 77pm2.61ii 157 . . 3  |-  ( ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B ) ) )
79783expib 1154 . 2  |-  ( W  e.  _V  ->  (
( A  e.  X  /\  B  e.  Y
)  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) ) )
80 ress0 13202 . . . 4  |-  ( (/)s  B )  =  (/)
81 reldmress 13194 . . . . . 6  |-  Rel  doms
8281ovprc1 5886 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  (/) )
8382oveq1d 5873 . . . 4  |-  ( -.  W  e.  _V  ->  ( ( Ws  A )s  B )  =  (
(/)s  B ) )
8481ovprc1 5886 . . . 4  |-  ( -.  W  e.  _V  ->  ( Ws  ( A  i^i  B
) )  =  (/) )
8580, 83, 843eqtr4a 2341 . . 3  |-  ( -.  W  e.  _V  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) )
8685a1d 22 . 2  |-  ( -.  W  e.  _V  ->  ( ( A  e.  X  /\  B  e.  Y
)  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) ) )
8779, 86pm2.61i 156 1  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   <.cop 3643   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146   Basecbs 13148   ↾s cress 13149
This theorem is referenced by:  ressabs  13206  xrge00  23311  esumpfinvallem  23442  lmhmlnmsplit  27185
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-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-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
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-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-recs 6388  df-rdg 6423  df-nn 9747  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155
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