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Theorem ressinbas 13525
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 2964 . 2  |-  ( A  e.  X  ->  A  e.  _V )
2 eqid 2436 . . . . . . 7  |-  ( Ws  A )  =  ( Ws  A )
3 ressid.1 . . . . . . 7  |-  B  =  ( Base `  W
)
42, 3ressid2 13517 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  W )
5 ssid 3367 . . . . . . . 8  |-  B  C_  B
6 incom 3533 . . . . . . . . 9  |-  ( A  i^i  B )  =  ( B  i^i  A
)
7 df-ss 3334 . . . . . . . . . 10  |-  ( B 
C_  A  <->  ( B  i^i  A )  =  B )
87biimpi 187 . . . . . . . . 9  |-  ( B 
C_  A  ->  ( B  i^i  A )  =  B )
96, 8syl5eq 2480 . . . . . . . 8  |-  ( B 
C_  A  ->  ( A  i^i  B )  =  B )
105, 9syl5sseqr 3397 . . . . . . 7  |-  ( B 
C_  A  ->  B  C_  ( A  i^i  B
) )
11 elex 2964 . . . . . . 7  |-  ( W  e.  _V  ->  W  e.  _V )
12 inex1g 4346 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  i^i  B )  e. 
_V )
13 eqid 2436 . . . . . . . 8  |-  ( Ws  ( A  i^i  B ) )  =  ( Ws  ( A  i^i  B ) )
1413, 3ressid2 13517 . . . . . . 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 1226 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  ( A  i^i  B ) )  =  W )
164, 15eqtr4d 2471 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
17163expb 1154 . . . 4  |-  ( ( B  C_  A  /\  ( W  e.  _V  /\  A  e.  _V )
)  ->  ( Ws  A
)  =  ( Ws  ( A  i^i  B ) ) )
18 inass 3551 . . . . . . . . 9  |-  ( ( A  i^i  B )  i^i  B )  =  ( A  i^i  ( B  i^i  B ) )
19 inidm 3550 . . . . . . . . . 10  |-  ( B  i^i  B )  =  B
2019ineq2i 3539 . . . . . . . . 9  |-  ( A  i^i  ( B  i^i  B ) )  =  ( A  i^i  B )
2118, 20eqtr2i 2457 . . . . . . . 8  |-  ( A  i^i  B )  =  ( ( A  i^i  B )  i^i  B )
2221opeq2i 3988 . . . . . . 7  |-  <. ( Base `  ndx ) ,  ( A  i^i  B
) >.  =  <. ( Base `  ndx ) ,  ( ( A  i^i  B )  i^i  B )
>.
2322oveq2i 6092 . . . . . 6  |-  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )  =  ( W sSet  <. (
Base `  ndx ) ,  ( ( A  i^i  B )  i^i  B )
>. )
242, 3ressval2 13518 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)
25 inss1 3561 . . . . . . . . 9  |-  ( A  i^i  B )  C_  A
26 sstr 3356 . . . . . . . . 9  |-  ( ( B  C_  ( A  i^i  B )  /\  ( A  i^i  B )  C_  A )  ->  B  C_  A )
2725, 26mpan2 653 . . . . . . . 8  |-  ( B 
C_  ( A  i^i  B )  ->  B  C_  A
)
2827con3i 129 . . . . . . 7  |-  ( -.  B  C_  A  ->  -.  B  C_  ( A  i^i  B ) )
2913, 3ressval2 13518 . . . . . . 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 1226 . . . . . 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 2494 . . . . 5  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
32313expb 1154 . . . 4  |-  ( ( -.  B  C_  A  /\  ( W  e.  _V  /\  A  e.  _V )
)  ->  ( Ws  A
)  =  ( Ws  ( A  i^i  B ) ) )
3317, 32pm2.61ian 766 . . 3  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
34 reldmress 13515 . . . . . 6  |-  Rel  doms
3534ovprc1 6109 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  (/) )
3634ovprc1 6109 . . . . 5  |-  ( -.  W  e.  _V  ->  ( Ws  ( A  i^i  B
) )  =  (/) )
3735, 36eqtr4d 2471 . . . 4  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
3837adantr 452 . . 3  |-  ( ( -.  W  e.  _V  /\  A  e.  _V )  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
3933, 38pm2.61ian 766 . 2  |-  ( A  e.  _V  ->  ( Ws  A )  =  ( Ws  ( A  i^i  B
) ) )
401, 39syl 16 1  |-  ( A  e.  X  ->  ( Ws  A )  =  ( 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 2956    i^i cin 3319    C_ wss 3320   (/)c0 3628   <.cop 3817   ` cfv 5454  (class class class)co 6081   ndxcnx 13466   sSet csts 13467   Basecbs 13469   ↾s cress 13470
This theorem is referenced by:  ressress  13526  rescabs  14033  resscat  14049  funcres2c  14098  ressffth  14135  cphsubrglem  19140  subofld  24245
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-ress 13476
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