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Theorem ressress 15199
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 763 . . . . . . . . 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 1015 . . . . . . . . 9  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  W  e.  _V )
3 simpr2 1016 . . . . . . . . 9  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  A  e.  X )
4 eqid 2453 . . . . . . . . . 10  |-  ( Ws  A )  =  ( Ws  A )
5 eqid 2453 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
64, 5ressval2 15190 . . . . . . . . 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 1269 . . . . . . . 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 3644 . . . . . . . . . . 11  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  =  ( A  i^i  ( B  i^i  ( Base `  W
) ) )
9 in12 3645 . . . . . . . . . . 11  |-  ( A  i^i  ( B  i^i  ( Base `  W )
) )  =  ( B  i^i  ( A  i^i  ( Base `  W
) ) )
108, 9eqtri 2475 . . . . . . . . . 10  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  =  ( B  i^i  ( A  i^i  ( Base `  W
) ) )
114, 5ressbas 15191 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  ( A  i^i  ( Base `  W
) )  =  (
Base `  ( Ws  A
) ) )
123, 11syl 17 . . . . . . . . . . 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 3636 . . . . . . . . . 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 2500 . . . . . . . . 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 4176 . . . . . . . 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 6313 . . . . . . 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 5880 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
1817inex2 4548 . . . . . . . 8  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  e.  _V
19 setsabs 15164 . . . . . . . 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 669 . . . . . . 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 2487 . . . . . 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 761 . . . . . . 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 6323 . . . . . . . 8  |-  ( Ws  A )  e.  _V
2423a1i 11 . . . . . . 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 1017 . . . . . . 7  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  B  e.  Y )
26 eqid 2453 . . . . . . . 8  |-  ( ( Ws  A )s  B )  =  ( ( Ws  A )s  B )
27 eqid 2453 . . . . . . . 8  |-  ( Base `  ( Ws  A ) )  =  ( Base `  ( Ws  A ) )
2826, 27ressval2 15190 . . . . . . 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 1269 . . . . . 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 3654 . . . . . . . . 9  |-  ( A  i^i  B )  C_  A
31 sstr 3442 . . . . . . . . 9  |-  ( ( ( Base `  W
)  C_  ( A  i^i  B )  /\  ( A  i^i  B )  C_  A )  ->  ( Base `  W )  C_  A )
3230, 31mpan2 678 . . . . . . . 8  |-  ( (
Base `  W )  C_  ( A  i^i  B
)  ->  ( Base `  W )  C_  A
)
331, 32nsyl 125 . . . . . . 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 4549 . . . . . . . 8  |-  ( A  e.  X  ->  ( A  i^i  B )  e. 
_V )
353, 34syl 17 . . . . . . 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 2453 . . . . . . . 8  |-  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( A  i^i  B ) )
3736, 5ressval2 15190 . . . . . . 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 1269 . . . . . 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 2497 . . . . 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 609 . . . 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 15189 . . . . . . . 8  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( Ws  A )  e.  _V  /\  B  e.  Y )  ->  (
( Ws  A )s  B )  =  ( Ws  A ) )
4223, 41mp3an2 1354 . . . . . . 7  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  B  e.  Y
)  ->  ( ( Ws  A )s  B )  =  ( Ws  A ) )
43423ad2antr3 1176 . . . . . 6  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  (
( Ws  A )s  B )  =  ( Ws  A ) )
44 in32 3646 . . . . . . . . 9  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  =  ( ( A  i^i  ( Base `  W ) )  i^i  B )
45 simpr2 1016 . . . . . . . . . . . 12  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  A  e.  X )
4645, 11syl 17 . . . . . . . . . . 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 459 . . . . . . . . . . 11  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( Base `  ( Ws  A ) )  C_  B )
4846, 47eqsstrd 3468 . . . . . . . . . 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 3420 . . . . . . . . . 10  |-  ( ( A  i^i  ( Base `  W ) )  C_  B 
<->  ( ( A  i^i  ( Base `  W )
)  i^i  B )  =  ( A  i^i  ( Base `  W )
) )
5048, 49sylib 200 . . . . . . . . 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 2500 . . . . . . . 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 6311 . . . . . . 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 15197 . . . . . . . 8  |-  ( A  e.  X  ->  ( Ws  A )  =  ( Ws  ( A  i^i  ( Base `  W ) ) ) )
5445, 53syl 17 . . . . . . 7  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  ( Base `  W ) ) ) )
555ressinbas 15197 . . . . . . . 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 2497 . . . . . 6  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
5843, 57eqtrd 2487 . . . . 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 436 . . . 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 15189 . . . . . . . 8  |-  ( ( ( Base `  W
)  C_  A  /\  W  e.  _V  /\  A  e.  X )  ->  ( Ws  A )  =  W )
61603adant3r3 1220 . . . . . . 7  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Ws  A
)  =  W )
6261oveq1d 6310 . . . . . 6  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( ( Ws  A )s  B )  =  ( Ws  B ) )
63 inss2 3655 . . . . . . . . . . 11  |-  ( B  i^i  ( Base `  W
) )  C_  ( Base `  W )
64 simpl 459 . . . . . . . . . . 11  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Base `  W )  C_  A
)
6563, 64syl5ss 3445 . . . . . . . . . 10  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( B  i^i  ( Base `  W
) )  C_  A
)
66 sseqin2 3653 . . . . . . . . . 10  |-  ( ( B  i^i  ( Base `  W ) )  C_  A 
<->  ( A  i^i  ( B  i^i  ( Base `  W
) ) )  =  ( B  i^i  ( Base `  W ) ) )
6765, 66sylib 200 . . . . . . . . 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 2500 . . . . . . . 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 6311 . . . . . . 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 1017 . . . . . . . 8  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  B  e.  Y )
715ressinbas 15197 . . . . . . . 8  |-  ( B  e.  Y  ->  ( Ws  B )  =  ( Ws  ( B  i^i  ( Base `  W ) ) ) )
7270, 71syl 17 . . . . . . 7  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Ws  B
)  =  ( Ws  ( B  i^i  ( Base `  W ) ) ) )
73 simpr2 1016 . . . . . . . 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 2497 . . . . . 6  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Ws  B
)  =  ( Ws  ( A  i^i  B ) ) )
7662, 75eqtrd 2487 . . . . 5  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) )
7776ex 436 . . . 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 169 . . 3  |-  ( ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B ) ) )
79783expib 1212 . 2  |-  ( W  e.  _V  ->  (
( A  e.  X  /\  B  e.  Y
)  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) ) )
80 ress0 15195 . . . 4  |-  ( (/)s  B )  =  (/)
81 reldmress 15187 . . . . . 6  |-  Rel  doms
8281ovprc1 6326 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  (/) )
8382oveq1d 6310 . . . 4  |-  ( -.  W  e.  _V  ->  ( ( Ws  A )s  B )  =  (
(/)s  B ) )
8481ovprc1 6326 . . . 4  |-  ( -.  W  e.  _V  ->  ( Ws  ( A  i^i  B
) )  =  (/) )
8580, 83, 843eqtr4a 2513 . . 3  |-  ( -.  W  e.  _V  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) )
8685a1d 26 . 2  |-  ( -.  W  e.  _V  ->  ( ( A  e.  X  /\  B  e.  Y
)  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) ) )
8779, 86pm2.61i 168 1  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   _Vcvv 3047    i^i cin 3405    C_ wss 3406   (/)c0 3733   <.cop 3976   ` cfv 5585  (class class class)co 6295   ndxcnx 15130   sSet csts 15131   Basecbs 15133   ↾s cress 15134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-i2m1 9612  ax-1ne0 9613  ax-rrecex 9616  ax-cnre 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-nn 10617  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140
This theorem is referenced by:  ressabs  15200  xrge00  28460  xrge0slmod  28619  esumpfinvallem  28907  lmhmlnmsplit  35957
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