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Theorem ssrnres 5116
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 3390 . . . . 5  |-  ( C  i^i  ( A  X.  B ) )  C_  ( A  X.  B
)
2 rnss 4907 . . . . 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 8 . . . 4  |-  ran  ( C  i^i  ( A  X.  B ) )  C_  ran  ( A  X.  B
)
4 rnxpss 5108 . . . 4  |-  ran  ( A  X.  B )  C_  B
53, 4sstri 3188 . . 3  |-  ran  ( C  i^i  ( A  X.  B ) )  C_  B
6 eqss 3194 . . 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 884 . 2  |-  ( ran  ( C  i^i  ( A  X.  B ) )  =  B  <->  B  C_  ran  ( C  i^i  ( A  X.  B ) ) )
8 ssid 3197 . . . . . . . 8  |-  A  C_  A
9 ssv 3198 . . . . . . . 8  |-  B  C_  _V
10 xpss12 4792 . . . . . . . 8  |-  ( ( A  C_  A  /\  B  C_  _V )  -> 
( A  X.  B
)  C_  ( A  X.  _V ) )
118, 9, 10mp2an 653 . . . . . . 7  |-  ( A  X.  B )  C_  ( A  X.  _V )
12 sslin 3395 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( A  X.  _V )  ->  ( C  i^i  ( A  X.  B ) )  C_  ( C  i^i  ( A  X.  _V ) ) )
1311, 12ax-mp 8 . . . . . 6  |-  ( C  i^i  ( A  X.  B ) )  C_  ( C  i^i  ( A  X.  _V ) )
14 df-res 4701 . . . . . 6  |-  ( C  |`  A )  =  ( C  i^i  ( A  X.  _V ) )
1513, 14sseqtr4i 3211 . . . . 5  |-  ( C  i^i  ( A  X.  B ) )  C_  ( C  |`  A )
16 rnss 4907 . . . . 5  |-  ( ( C  i^i  ( A  X.  B ) ) 
C_  ( C  |`  A )  ->  ran  ( C  i^i  ( A  X.  B ) ) 
C_  ran  ( C  |`  A ) )
1715, 16ax-mp 8 . . . 4  |-  ran  ( C  i^i  ( A  X.  B ) )  C_  ran  ( C  |`  A )
18 sstr 3187 . . . 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 652 . . 3  |-  ( B 
C_  ran  ( C  i^i  ( A  X.  B
) )  ->  B  C_ 
ran  ( C  |`  A ) )
20 ssel 3174 . . . . . . 7  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  -> 
y  e.  ran  ( C  |`  A ) ) )
21 vex 2791 . . . . . . . 8  |-  y  e. 
_V
2221elrn2 4918 . . . . . . 7  |-  ( y  e.  ran  ( C  |`  A )  <->  E. x <. x ,  y >.  e.  ( C  |`  A ) )
2320, 22syl6ib 217 . . . . . 6  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  ->  E. x <. x ,  y
>.  e.  ( C  |`  A ) ) )
2423ancrd 537 . . . . 5  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  -> 
( E. x <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
) ) )
2521elrn2 4918 . . . . . 6  |-  ( y  e.  ran  ( C  i^i  ( A  X.  B ) )  <->  E. x <. x ,  y >.  e.  ( C  i^i  ( A  X.  B ) ) )
26 elin 3358 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  ( C  i^i  ( A  X.  B ) )  <-> 
( <. x ,  y
>.  e.  C  /\  <. x ,  y >.  e.  ( A  X.  B ) ) )
27 opelxp 4719 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
2827anbi2i 675 . . . . . . . 8  |-  ( (
<. x ,  y >.  e.  C  /\  <. x ,  y >.  e.  ( A  X.  B ) )  <->  ( <. x ,  y >.  e.  C  /\  ( x  e.  A  /\  y  e.  B
) ) )
2921opelres 4960 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( C  |`  A )  <-> 
( <. x ,  y
>.  e.  C  /\  x  e.  A ) )
3029anbi1i 676 . . . . . . . . 9  |-  ( (
<. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
)  <->  ( ( <.
x ,  y >.  e.  C  /\  x  e.  A )  /\  y  e.  B ) )
31 anass 630 . . . . . . . . 9  |-  ( ( ( <. x ,  y
>.  e.  C  /\  x  e.  A )  /\  y  e.  B )  <->  ( <. x ,  y >.  e.  C  /\  ( x  e.  A  /\  y  e.  B
) ) )
3230, 31bitr2i 241 . . . . . . . 8  |-  ( (
<. x ,  y >.  e.  C  /\  (
x  e.  A  /\  y  e.  B )
)  <->  ( <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
) )
3326, 28, 323bitri 262 . . . . . . 7  |-  ( <.
x ,  y >.  e.  ( C  i^i  ( A  X.  B ) )  <-> 
( <. x ,  y
>.  e.  ( C  |`  A )  /\  y  e.  B ) )
3433exbii 1569 . . . . . 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 1842 . . . . . 6  |-  ( E. x ( <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
)  <->  ( E. x <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
) )
3625, 34, 353bitri 262 . . . . 5  |-  ( y  e.  ran  ( C  i^i  ( A  X.  B ) )  <->  ( E. x <. x ,  y
>.  e.  ( C  |`  A )  /\  y  e.  B ) )
3724, 36syl6ibr 218 . . . 4  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  -> 
y  e.  ran  ( C  i^i  ( A  X.  B ) ) ) )
3837ssrdv 3185 . . 3  |-  ( B 
C_  ran  ( C  |`  A )  ->  B  C_ 
ran  ( C  i^i  ( A  X.  B
) ) )
3919, 38impbii 180 . 2  |-  ( B 
C_  ran  ( C  i^i  ( A  X.  B
) )  <->  B  C_  ran  ( C  |`  A ) )
407, 39bitr2i 241 1  |-  ( B 
C_  ran  ( C  |`  A )  <->  ran  ( C  i^i  ( A  X.  B ) )  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   <.cop 3643    X. cxp 4687   ran crn 4690    |` cres 4691
This theorem is referenced by:  rninxp  5117
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701
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