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Theorem ressress 14548
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 754 . . . . . . . . 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 1002 . . . . . . . . 9  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  W  e.  _V )
3 simpr2 1003 . . . . . . . . 9  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  A  e.  X )
4 eqid 2467 . . . . . . . . . 10  |-  ( Ws  A )  =  ( Ws  A )
5 eqid 2467 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
64, 5ressval2 14540 . . . . . . . . 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 1228 . . . . . . . 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 3708 . . . . . . . . . . 11  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  =  ( A  i^i  ( B  i^i  ( Base `  W
) ) )
9 in12 3709 . . . . . . . . . . 11  |-  ( A  i^i  ( B  i^i  ( Base `  W )
) )  =  ( B  i^i  ( A  i^i  ( Base `  W
) ) )
108, 9eqtri 2496 . . . . . . . . . 10  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  =  ( B  i^i  ( A  i^i  ( Base `  W
) ) )
114, 5ressbas 14541 . . . . . . . . . . . 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 3700 . . . . . . . . . 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 2521 . . . . . . . . 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 4220 . . . . . . . 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 6300 . . . . . . 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 5874 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
1817inex2 4589 . . . . . . . 8  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  e.  _V
19 setsabs 14515 . . . . . . . 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 662 . . . . . . 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 2508 . . . . . 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 753 . . . . . . 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 6307 . . . . . . . 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 1004 . . . . . . 7  |-  ( ( ( -.  ( Base `  ( Ws  A ) )  C_  B  /\  -.  ( Base `  W )  C_  A
)  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y ) )  ->  B  e.  Y )
26 eqid 2467 . . . . . . . 8  |-  ( ( Ws  A )s  B )  =  ( ( Ws  A )s  B )
27 eqid 2467 . . . . . . . 8  |-  ( Base `  ( Ws  A ) )  =  ( Base `  ( Ws  A ) )
2826, 27ressval2 14540 . . . . . . 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 1228 . . . . . 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 3718 . . . . . . . . 9  |-  ( A  i^i  B )  C_  A
31 sstr 3512 . . . . . . . . 9  |-  ( ( ( Base `  W
)  C_  ( A  i^i  B )  /\  ( A  i^i  B )  C_  A )  ->  ( Base `  W )  C_  A )
3230, 31mpan2 671 . . . . . . . 8  |-  ( (
Base `  W )  C_  ( A  i^i  B
)  ->  ( Base `  W )  C_  A
)
331, 32nsyl 121 . . . . . . 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 4590 . . . . . . . 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 2467 . . . . . . . 8  |-  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( A  i^i  B ) )
3736, 5ressval2 14540 . . . . . . 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 1228 . . . . . 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 2518 . . . . 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 604 . . . 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 14539 . . . . . . . 8  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( Ws  A )  e.  _V  /\  B  e.  Y )  ->  (
( Ws  A )s  B )  =  ( Ws  A ) )
4223, 41mp3an2 1312 . . . . . . 7  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  B  e.  Y
)  ->  ( ( Ws  A )s  B )  =  ( Ws  A ) )
43423ad2antr3 1163 . . . . . 6  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  (
( Ws  A )s  B )  =  ( Ws  A ) )
44 in32 3710 . . . . . . . . 9  |-  ( ( A  i^i  B )  i^i  ( Base `  W
) )  =  ( ( A  i^i  ( Base `  W ) )  i^i  B )
45 simpr2 1003 . . . . . . . . . . . 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 457 . . . . . . . . . . 11  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( Base `  ( Ws  A ) )  C_  B )
4846, 47eqsstrd 3538 . . . . . . . . . 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 3490 . . . . . . . . . 10  |-  ( ( A  i^i  ( Base `  W ) )  C_  B 
<->  ( ( A  i^i  ( Base `  W )
)  i^i  B )  =  ( A  i^i  ( Base `  W )
) )
5048, 49sylib 196 . . . . . . . . 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 2521 . . . . . . . 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 6298 . . . . . . 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 14547 . . . . . . . 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 14547 . . . . . . . 8  |-  ( ( A  i^i  B )  e.  _V  ->  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( ( A  i^i  B
)  i^i  ( Base `  W ) ) ) )
5645, 34, 553syl 20 . . . . . . 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 2518 . . . . . 6  |-  ( ( ( Base `  ( Ws  A ) )  C_  B  /\  ( W  e. 
_V  /\  A  e.  X  /\  B  e.  Y
) )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
5843, 57eqtrd 2508 . . . . 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 434 . . . 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 14539 . . . . . . . 8  |-  ( ( ( Base `  W
)  C_  A  /\  W  e.  _V  /\  A  e.  X )  ->  ( Ws  A )  =  W )
61603adant3r3 1207 . . . . . . 7  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Ws  A
)  =  W )
6261oveq1d 6297 . . . . . 6  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( ( Ws  A )s  B )  =  ( Ws  B ) )
63 inss2 3719 . . . . . . . . . . 11  |-  ( B  i^i  ( Base `  W
) )  C_  ( Base `  W )
64 simpl 457 . . . . . . . . . . 11  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Base `  W )  C_  A
)
6563, 64syl5ss 3515 . . . . . . . . . 10  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( B  i^i  ( Base `  W
) )  C_  A
)
66 sseqin2 3717 . . . . . . . . . 10  |-  ( ( B  i^i  ( Base `  W ) )  C_  A 
<->  ( A  i^i  ( B  i^i  ( Base `  W
) ) )  =  ( B  i^i  ( Base `  W ) ) )
6765, 66sylib 196 . . . . . . . . 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 2521 . . . . . . . 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 6298 . . . . . . 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 1004 . . . . . . . 8  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  B  e.  Y )
715ressinbas 14547 . . . . . . . 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 1003 . . . . . . . 8  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  A  e.  X )
7473, 34, 553syl 20 . . . . . . 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 2518 . . . . . 6  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( Ws  B
)  =  ( Ws  ( A  i^i  B ) ) )
7662, 75eqtrd 2508 . . . . 5  |-  ( ( ( Base `  W
)  C_  A  /\  ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) )
7776ex 434 . . . 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 165 . . 3  |-  ( ( W  e.  _V  /\  A  e.  X  /\  B  e.  Y )  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B ) ) )
79783expib 1199 . 2  |-  ( W  e.  _V  ->  (
( A  e.  X  /\  B  e.  Y
)  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) ) )
80 ress0 14545 . . . 4  |-  ( (/)s  B )  =  (/)
81 reldmress 14537 . . . . . 6  |-  Rel  doms
8281ovprc1 6310 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  (/) )
8382oveq1d 6297 . . . 4  |-  ( -.  W  e.  _V  ->  ( ( Ws  A )s  B )  =  (
(/)s  B ) )
8481ovprc1 6310 . . . 4  |-  ( -.  W  e.  _V  ->  ( Ws  ( A  i^i  B
) )  =  (/) )
8580, 83, 843eqtr4a 2534 . . 3  |-  ( -.  W  e.  _V  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) )
8685a1d 25 . 2  |-  ( -.  W  e.  _V  ->  ( ( A  e.  X  /\  B  e.  Y
)  ->  ( ( Ws  A )s  B )  =  ( Ws  ( A  i^i  B
) ) ) )
8779, 86pm2.61i 164 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   <.cop 4033   ` cfv 5586  (class class class)co 6282   ndxcnx 14483   sSet csts 14484   Basecbs 14486   ↾s cress 14487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-i2m1 9556  ax-1ne0 9557  ax-rrecex 9560  ax-cnre 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-nn 10533  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493
This theorem is referenced by:  ressabs  14549  xrge00  27336  xrge0slmod  27497  esumpfinvallem  27720  lmhmlnmsplit  30637
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