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Theorem ssrnres 4791
Description: Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)
Assertion
Ref Expression
ssrnres  |-  ( B 
C_  ran  ( C  |`  A )  <->  ran  ( C  i^i  ( A  X.  B ) )  =  B )

Proof of Theorem ssrnres
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 3186 . . . . 5  |-  ( C  i^i  ( A  X.  B ) )  C_  ( A  X.  B
)
2 rnss 4592 . . . . 5  |-  ( ( C  i^i  ( A  X.  B ) ) 
C_  ( A  X.  B )  ->  ran  ( C  i^i  ( A  X.  B ) ) 
C_  ran  ( A  X.  B ) )
31, 2ax-mp 7 . . . 4  |-  ran  ( C  i^i  ( A  X.  B ) )  C_  ran  ( A  X.  B
)
4 rnxpss 4782 . . . 4  |-  ran  ( A  X.  B )  C_  B
53, 4sstri 2982 . . 3  |-  ran  ( C  i^i  ( A  X.  B ) )  C_  B
6 eqss 2988 . . 3  |-  ( ran  ( C  i^i  ( A  X.  B ) )  =  B  <->  ( ran  ( C  i^i  ( A  X.  B ) ) 
C_  B  /\  B  C_ 
ran  ( C  i^i  ( A  X.  B
) ) ) )
75, 6mpbiran 858 . 2  |-  ( ran  ( C  i^i  ( A  X.  B ) )  =  B  <->  B  C_  ran  ( C  i^i  ( A  X.  B ) ) )
8 ssid 2992 . . . . . . . 8  |-  A  C_  A
9 ssv 2993 . . . . . . . 8  |-  B  C_  _V
10 xpss12 4473 . . . . . . . 8  |-  ( ( A  C_  A  /\  B  C_  _V )  -> 
( A  X.  B
)  C_  ( A  X.  _V ) )
118, 9, 10mp2an 410 . . . . . . 7  |-  ( A  X.  B )  C_  ( A  X.  _V )
12 sslin 3191 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( A  X.  _V )  ->  ( C  i^i  ( A  X.  B ) )  C_  ( C  i^i  ( A  X.  _V ) ) )
1311, 12ax-mp 7 . . . . . 6  |-  ( C  i^i  ( A  X.  B ) )  C_  ( C  i^i  ( A  X.  _V ) )
14 df-res 4385 . . . . . 6  |-  ( C  |`  A )  =  ( C  i^i  ( A  X.  _V ) )
1513, 14sseqtr4i 3006 . . . . 5  |-  ( C  i^i  ( A  X.  B ) )  C_  ( C  |`  A )
16 rnss 4592 . . . . 5  |-  ( ( C  i^i  ( A  X.  B ) ) 
C_  ( C  |`  A )  ->  ran  ( C  i^i  ( A  X.  B ) ) 
C_  ran  ( C  |`  A ) )
1715, 16ax-mp 7 . . . 4  |-  ran  ( C  i^i  ( A  X.  B ) )  C_  ran  ( C  |`  A )
18 sstr 2981 . . . 4  |-  ( ( B  C_  ran  ( C  i^i  ( A  X.  B ) )  /\  ran  ( C  i^i  ( A  X.  B ) ) 
C_  ran  ( C  |`  A ) )  ->  B  C_  ran  ( C  |`  A ) )
1917, 18mpan2 409 . . 3  |-  ( B 
C_  ran  ( C  i^i  ( A  X.  B
) )  ->  B  C_ 
ran  ( C  |`  A ) )
20 ssel 2967 . . . . . . 7  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  -> 
y  e.  ran  ( C  |`  A ) ) )
21 vex 2577 . . . . . . . 8  |-  y  e. 
_V
2221elrn2 4604 . . . . . . 7  |-  ( y  e.  ran  ( C  |`  A )  <->  E. x <. x ,  y >.  e.  ( C  |`  A ) )
2320, 22syl6ib 154 . . . . . 6  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  ->  E. x <. x ,  y
>.  e.  ( C  |`  A ) ) )
2423ancrd 313 . . . . 5  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  -> 
( E. x <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
) ) )
2521elrn2 4604 . . . . . 6  |-  ( y  e.  ran  ( C  i^i  ( A  X.  B ) )  <->  E. x <. x ,  y >.  e.  ( C  i^i  ( A  X.  B ) ) )
26 elin 3154 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  ( C  i^i  ( A  X.  B ) )  <-> 
( <. x ,  y
>.  e.  C  /\  <. x ,  y >.  e.  ( A  X.  B ) ) )
27 opelxp 4402 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
2827anbi2i 438 . . . . . . . 8  |-  ( (
<. x ,  y >.  e.  C  /\  <. x ,  y >.  e.  ( A  X.  B ) )  <->  ( <. x ,  y >.  e.  C  /\  ( x  e.  A  /\  y  e.  B
) ) )
2921opelres 4645 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( C  |`  A )  <-> 
( <. x ,  y
>.  e.  C  /\  x  e.  A ) )
3029anbi1i 439 . . . . . . . . 9  |-  ( (
<. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
)  <->  ( ( <.
x ,  y >.  e.  C  /\  x  e.  A )  /\  y  e.  B ) )
31 anass 387 . . . . . . . . 9  |-  ( ( ( <. x ,  y
>.  e.  C  /\  x  e.  A )  /\  y  e.  B )  <->  ( <. x ,  y >.  e.  C  /\  ( x  e.  A  /\  y  e.  B
) ) )
3230, 31bitr2i 178 . . . . . . . 8  |-  ( (
<. x ,  y >.  e.  C  /\  (
x  e.  A  /\  y  e.  B )
)  <->  ( <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
) )
3326, 28, 323bitri 199 . . . . . . 7  |-  ( <.
x ,  y >.  e.  ( C  i^i  ( A  X.  B ) )  <-> 
( <. x ,  y
>.  e.  ( C  |`  A )  /\  y  e.  B ) )
3433exbii 1512 . . . . . 6  |-  ( E. x <. x ,  y
>.  e.  ( C  i^i  ( A  X.  B
) )  <->  E. x
( <. x ,  y
>.  e.  ( C  |`  A )  /\  y  e.  B ) )
35 19.41v 1798 . . . . . 6  |-  ( E. x ( <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
)  <->  ( E. x <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
) )
3625, 34, 353bitri 199 . . . . 5  |-  ( y  e.  ran  ( C  i^i  ( A  X.  B ) )  <->  ( E. x <. x ,  y
>.  e.  ( C  |`  A )  /\  y  e.  B ) )
3724, 36syl6ibr 155 . . . 4  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  -> 
y  e.  ran  ( C  i^i  ( A  X.  B ) ) ) )
3837ssrdv 2979 . . 3  |-  ( B 
C_  ran  ( C  |`  A )  ->  B  C_ 
ran  ( C  i^i  ( A  X.  B
) ) )
3919, 38impbii 121 . 2  |-  ( B 
C_  ran  ( C  i^i  ( A  X.  B
) )  <->  B  C_  ran  ( C  |`  A ) )
407, 39bitr2i 178 1  |-  ( B 
C_  ran  ( C  |`  A )  <->  ran  ( C  i^i  ( A  X.  B ) )  =  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574    i^i cin 2944    C_ wss 2945   <.cop 3406    X. cxp 4371   ran crn 4374    |` cres 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385
This theorem is referenced by:  rninxp  4792
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