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Theorem ressinbas 13220
Description: Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypothesis
Ref Expression
ressid.1  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
ressinbas  |-  ( A  e.  X  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )

Proof of Theorem ressinbas
StepHypRef Expression
1 elex 2809 . 2  |-  ( A  e.  X  ->  A  e.  _V )
2 eqid 2296 . . . . . . 7  |-  ( Ws  A )  =  ( Ws  A )
3 ressid.1 . . . . . . 7  |-  B  =  ( Base `  W
)
42, 3ressid2 13212 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  W )
5 ssid 3210 . . . . . . . 8  |-  B  C_  B
6 incom 3374 . . . . . . . . 9  |-  ( A  i^i  B )  =  ( B  i^i  A
)
7 df-ss 3179 . . . . . . . . . 10  |-  ( B 
C_  A  <->  ( B  i^i  A )  =  B )
87biimpi 186 . . . . . . . . 9  |-  ( B 
C_  A  ->  ( B  i^i  A )  =  B )
96, 8syl5eq 2340 . . . . . . . 8  |-  ( B 
C_  A  ->  ( A  i^i  B )  =  B )
105, 9syl5sseqr 3240 . . . . . . 7  |-  ( B 
C_  A  ->  B  C_  ( A  i^i  B
) )
11 elex 2809 . . . . . . 7  |-  ( W  e.  _V  ->  W  e.  _V )
12 inex1g 4173 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  i^i  B )  e. 
_V )
13 eqid 2296 . . . . . . . 8  |-  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( A  i^i  B ) )
1413, 3ressid2 13212 . . . . . . 7  |-  ( ( B  C_  ( A  i^i  B )  /\  W  e.  _V  /\  ( A  i^i  B )  e. 
_V )  ->  ( Ws  ( A  i^i  B ) )  =  W )
1510, 11, 12, 14syl3an 1224 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  ( A  i^i  B ) )  =  W )
164, 15eqtr4d 2331 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
17163expb 1152 . . . 4  |-  ( ( B  C_  A  /\  ( W  e.  _V  /\  A  e.  _V )
)  ->  ( Ws  A
)  =  ( Ws  ( A  i^i  B ) ) )
18 inass 3392 . . . . . . . . 9  |-  ( ( A  i^i  B )  i^i  B )  =  ( A  i^i  ( B  i^i  B ) )
19 inidm 3391 . . . . . . . . . 10  |-  ( B  i^i  B )  =  B
2019ineq2i 3380 . . . . . . . . 9  |-  ( A  i^i  ( B  i^i  B ) )  =  ( A  i^i  B )
2118, 20eqtr2i 2317 . . . . . . . 8  |-  ( A  i^i  B )  =  ( ( A  i^i  B )  i^i  B )
2221opeq2i 3816 . . . . . . 7  |-  <. ( Base `  ndx ) ,  ( A  i^i  B
) >.  =  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  B )
>.
2322oveq2i 5885 . . . . . 6  |-  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )  =  ( W sSet  <. (
Base `  ndx ) ,  ( ( A  i^i  B )  i^i  B )
>. )
242, 3ressval2 13213 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)
25 inss1 3402 . . . . . . . . 9  |-  ( A  i^i  B )  C_  A
26 sstr 3200 . . . . . . . . 9  |-  ( ( B  C_  ( A  i^i  B )  /\  ( A  i^i  B )  C_  A )  ->  B  C_  A )
2725, 26mpan2 652 . . . . . . . 8  |-  ( B 
C_  ( A  i^i  B )  ->  B  C_  A
)
2827con3i 127 . . . . . . 7  |-  ( -.  B  C_  A  ->  -.  B  C_  ( A  i^i  B ) )
2913, 3ressval2 13213 . . . . . . 7  |-  ( ( -.  B  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 
B ) >. )
)
3028, 11, 12, 29syl3an 1224 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  ( A  i^i  B ) )  =  ( W sSet  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i 
B ) >. )
)
3123, 24, 303eqtr4a 2354 . . . . 5  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
32313expb 1152 . . . 4  |-  ( ( -.  B  C_  A  /\  ( W  e.  _V  /\  A  e.  _V )
)  ->  ( Ws  A
)  =  ( Ws  ( A  i^i  B ) ) )
3317, 32pm2.61ian 765 . . 3  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
34 reldmress 13210 . . . . . 6  |-  Rel  doms
3534ovprc1 5902 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  (/) )
3634ovprc1 5902 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  ( A  i^i  B
) )  =  (/) )
3735, 36eqtr4d 2331 . . . 4  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
3837adantr 451 . . 3  |-  ( ( -.  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
3933, 38pm2.61ian 765 . 2  |-  ( A  e.  _V  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
401, 39syl 15 1  |-  ( A  e.  X  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   <.cop 3656   ` cfv 5271  (class class class)co 5874   ndxcnx 13161   sSet csts 13162   Basecbs 13164   ↾s cress 13165
This theorem is referenced by:  ressress  13221  rescabs  13726  resscat  13742  funcres2c  13791  ressffth  13828  cphsubrglem  18629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-ress 13171
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