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Theorem ressress 13518
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 732 . . . . . . . . 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 963 . . . . . . . . 9  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  W  e.  _V )
3 simpr2 964 . . . . . . . . 9  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  A  e.  X )
4 eqid 2435 . . . . . . . . . 10  |-  ( Ws  A )  =  ( Ws  A )
5 eqid 2435 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
64, 5ressval2 13510 . . . . . . . . 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 1184 . . . . . . . 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 3543 . . . . . . . . . . 11  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  =  ( A  i^i  ( B  i^i  ( Base `  W
) ) )
9 in12 3544 . . . . . . . . . . 11  |-  ( A  i^i  ( B  i^i  ( Base `  W )
) )  =  ( B  i^i  ( A  i^i  ( Base `  W
) ) )
108, 9eqtri 2455 . . . . . . . . . 10  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  =  ( B  i^i  ( A  i^i  ( Base `  W
) ) )
114, 5ressbas 13511 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  ( A  i^i  ( Base `  W
) )  =  (
Base `  ( Ws  A
) ) )
123, 11syl 16 . . . . . . . . . . 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 3534 . . . . . . . . . 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 2480 . . . . . . . . 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 3983 . . . . . . . 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 6091 . . . . . . 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 5734 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
1817inex2 4337 . . . . . . . 8  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  e.  _V
19 setsabs 13488 . . . . . . . 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 644 . . . . . . 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 2467 . . . . . 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 731 . . . . . . 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 6098 . . . . . . . 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 965 . . . . . . 7  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  B  e.  Y )
26 eqid 2435 . . . . . . . 8  |-  ( ( Ws  A )s  B )  =  ( ( Ws  A )s  B )
27 eqid 2435 . . . . . . . 8  |-  ( Base `  ( Ws  A ) )  =  ( Base `  ( Ws  A ) )
2826, 27ressval2 13510 . . . . . . 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 1184 . . . . . 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 3553 . . . . . . . . 9  |-  ( A  i^i  B )  C_  A
31 sstr 3348 . . . . . . . . 9  |-  ( ( ( Base `  W
)  C_  ( A  i^i  B )  /\  ( A  i^i  B )  C_  A )  ->  ( Base `  W )  C_  A )
3230, 31mpan2 653 . . . . . . . 8  |-  ( (
Base `  W )  C_  ( A  i^i  B
)  ->  ( Base `  W )  C_  A
)
331, 32nsyl 115 . . . . . . 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 4338 . . . . . . . 8  |-  ( A  e.  X  ->  ( A  i^i  B )  e. 
_V )
353, 34syl 16 . . . . . . 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 2435 . . . . . . . 8  |-  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( A  i^i  B ) )
3736, 5ressval2 13510 . . . . . . 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 1184 . . . . . 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 2477 . . . . 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 588 . . . 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 13509 . . . . . . . 8  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( Ws  A )  e.  _V  /\  B  e.  Y )  ->  (
( Ws  A )s  B )  =  ( Ws  A ) )
4223, 41mp3an2 1267 . . . . . . 7  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  B  e.  Y
)  ->  ( ( Ws  A )s  B )  =  ( Ws  A ) )
43423ad2antr3 1124 . . . . . 6  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  (
( Ws  A )s  B )  =  ( Ws  A ) )
44 in32 3545 . . . . . . . . 9  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  =  ( ( A  i^i  ( Base `  W ) )  i^i  B )
45 simpr2 964 . . . . . . . . . . . 12  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  A  e.  X )
4645, 11syl 16 . . . . . . . . . . 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 444 . . . . . . . . . . 11  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( Base `  ( Ws  A ) )  C_  B )
4846, 47eqsstrd 3374 . . . . . . . . . 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 3326 . . . . . . . . . 10  |-  ( ( A  i^i  ( Base `  W ) )  C_  B 
<->  ( ( A  i^i  ( Base `  W )
)  i^i  B )  =  ( A  i^i  ( Base `  W )
) )
5048, 49sylib 189 . . . . . . . . 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 2480 . . . . . . . 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 6089 . . . . . . 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 13517 . . . . . . . 8  |-  ( A  e.  X  ->  ( Ws  A )  =  ( Ws  ( A  i^i  ( Base `  W ) ) ) )
5445, 53syl 16 . . . . . . 7  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  ( Base `  W ) ) ) )
555ressinbas 13517 . . . . . . . 8  |-  ( ( A  i^i  B )  e.  _V  ->  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( ( A  i^i  B
)  i^i  ( Base `  W ) ) ) )
5645, 34, 553syl 19 . . . . . . 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 2477 . . . . . 6  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
5843, 57eqtrd 2467 . . . . 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 424 . . . 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 13509 . . . . . . . 8  |-  ( ( ( Base `  W
)  C_  A  /\  W  e.  _V  /\  A  e.  X )  ->  ( Ws  A )  =  W )
61603adant3r3 1164 . . . . . . 7  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Ws  A
)  =  W )
6261oveq1d 6088 . . . . . 6  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( ( Ws  A )s  B )  =  ( Ws  B ) )
63 inss2 3554 . . . . . . . . . . 11  |-  ( B  i^i  ( Base `  W
) )  C_  ( Base `  W )
64 simpl 444 . . . . . . . . . . 11  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Base `  W )  C_  A
)
6563, 64syl5ss 3351 . . . . . . . . . 10  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( B  i^i  ( Base `  W
) )  C_  A
)
66 sseqin2 3552 . . . . . . . . . 10  |-  ( ( B  i^i  ( Base `  W ) )  C_  A 
<->  ( A  i^i  ( B  i^i  ( Base `  W
) ) )  =  ( B  i^i  ( Base `  W ) ) )
6765, 66sylib 189 . . . . . . . . 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 2480 . . . . . . . 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 6089 . . . . . . 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 965 . . . . . . . 8  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  B  e.  Y )
715ressinbas 13517 . . . . . . . 8  |-  ( B  e.  Y  ->  ( Ws  B )  =  ( Ws  ( B  i^i  ( Base `  W ) ) ) )
7270, 71syl 16 . . . . . . 7  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Ws  B
)  =  ( Ws  ( B  i^i  ( Base `  W ) ) ) )
73 simpr2 964 . . . . . . . 8  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  A  e.  X )
7473, 34, 553syl 19 . . . . . . 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 2477 . . . . . 6  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Ws  B
)  =  ( Ws  ( A  i^i  B ) ) )
7662, 75eqtrd 2467 . . . . 5  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) )
7776ex 424 . . . 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 159 . . 3  |-  ( ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B ) ) )
79783expib 1156 . 2  |-  ( W  e.  _V  ->  (
( A  e.  X  /\  B  e.  Y
)  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) ) )
80 ress0 13515 . . . 4  |-  ( (/)s  B )  =  (/)
81 reldmress 13507 . . . . . 6  |-  Rel  doms
8281ovprc1 6101 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  (/) )
8382oveq1d 6088 . . . 4  |-  ( -.  W  e.  _V  ->  ( ( Ws  A )s  B )  =  (
(/)s  B ) )
8481ovprc1 6101 . . . 4  |-  ( -.  W  e.  _V  ->  ( Ws  ( A  i^i  B
) )  =  (/) )
8580, 83, 843eqtr4a 2493 . . 3  |-  ( -.  W  e.  _V  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) )
8685a1d 23 . 2  |-  ( -.  W  e.  _V  ->  ( ( A  e.  X  /\  B  e.  Y
)  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) ) )
8779, 86pm2.61i 158 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620   <.cop 3809   ` cfv 5446  (class class class)co 6073   ndxcnx 13458   sSet csts 13459   Basecbs 13461   ↾s cress 13462
This theorem is referenced by:  ressabs  13519  xrge00  24200  esumpfinvallem  24456  lmhmlnmsplit  27143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rrecex 9054  ax-cnre 9055
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-nn 9993  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468
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