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Theorem ssrnres 5029
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 3328 . . . . 5  |-  ( C  i^i  ( A  X.  B ) )  C_  ( A  X.  B
)
2 rnss 4817 . . . . 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 5 . . . 4  |-  ran  ( C  i^i  ( A  X.  B ) )  C_  ran  ( A  X.  B
)
4 rnxpss 5018 . . . 4  |-  ran  ( A  X.  B )  C_  B
53, 4sstri 3137 . . 3  |-  ran  ( C  i^i  ( A  X.  B ) )  C_  B
6 eqss 3143 . . 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 925 . 2  |-  ( ran  ( C  i^i  ( A  X.  B ) )  =  B  <->  B  C_  ran  ( C  i^i  ( A  X.  B ) ) )
8 ssid 3148 . . . . . . . 8  |-  A  C_  A
9 ssv 3150 . . . . . . . 8  |-  B  C_  _V
10 xpss12 4694 . . . . . . . 8  |-  ( ( A  C_  A  /\  B  C_  _V )  -> 
( A  X.  B
)  C_  ( A  X.  _V ) )
118, 9, 10mp2an 423 . . . . . . 7  |-  ( A  X.  B )  C_  ( A  X.  _V )
12 sslin 3333 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( A  X.  _V )  ->  ( C  i^i  ( A  X.  B ) )  C_  ( C  i^i  ( A  X.  _V ) ) )
1311, 12ax-mp 5 . . . . . 6  |-  ( C  i^i  ( A  X.  B ) )  C_  ( C  i^i  ( A  X.  _V ) )
14 df-res 4599 . . . . . 6  |-  ( C  |`  A )  =  ( C  i^i  ( A  X.  _V ) )
1513, 14sseqtrri 3163 . . . . 5  |-  ( C  i^i  ( A  X.  B ) )  C_  ( C  |`  A )
16 rnss 4817 . . . . 5  |-  ( ( C  i^i  ( A  X.  B ) ) 
C_  ( C  |`  A )  ->  ran  ( C  i^i  ( A  X.  B ) ) 
C_  ran  ( C  |`  A ) )
1715, 16ax-mp 5 . . . 4  |-  ran  ( C  i^i  ( A  X.  B ) )  C_  ran  ( C  |`  A )
18 sstr 3136 . . . 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 422 . . 3  |-  ( B 
C_  ran  ( C  i^i  ( A  X.  B
) )  ->  B  C_ 
ran  ( C  |`  A ) )
20 ssel 3122 . . . . . . 7  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  -> 
y  e.  ran  ( C  |`  A ) ) )
21 vex 2715 . . . . . . . 8  |-  y  e. 
_V
2221elrn2 4829 . . . . . . 7  |-  ( y  e.  ran  ( C  |`  A )  <->  E. x <. x ,  y >.  e.  ( C  |`  A ) )
2320, 22syl6ib 160 . . . . . 6  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  ->  E. x <. x ,  y
>.  e.  ( C  |`  A ) ) )
2423ancrd 324 . . . . 5  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  -> 
( E. x <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
) ) )
2521elrn2 4829 . . . . . 6  |-  ( y  e.  ran  ( C  i^i  ( A  X.  B ) )  <->  E. x <. x ,  y >.  e.  ( C  i^i  ( A  X.  B ) ) )
26 elin 3290 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  ( C  i^i  ( A  X.  B ) )  <-> 
( <. x ,  y
>.  e.  C  /\  <. x ,  y >.  e.  ( A  X.  B ) ) )
27 opelxp 4617 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
2827anbi2i 453 . . . . . . . 8  |-  ( (
<. x ,  y >.  e.  C  /\  <. x ,  y >.  e.  ( A  X.  B ) )  <->  ( <. x ,  y >.  e.  C  /\  ( x  e.  A  /\  y  e.  B
) ) )
2921opelres 4872 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( C  |`  A )  <-> 
( <. x ,  y
>.  e.  C  /\  x  e.  A ) )
3029anbi1i 454 . . . . . . . . 9  |-  ( (
<. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
)  <->  ( ( <.
x ,  y >.  e.  C  /\  x  e.  A )  /\  y  e.  B ) )
31 anass 399 . . . . . . . . 9  |-  ( ( ( <. x ,  y
>.  e.  C  /\  x  e.  A )  /\  y  e.  B )  <->  ( <. x ,  y >.  e.  C  /\  ( x  e.  A  /\  y  e.  B
) ) )
3230, 31bitr2i 184 . . . . . . . 8  |-  ( (
<. x ,  y >.  e.  C  /\  (
x  e.  A  /\  y  e.  B )
)  <->  ( <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
) )
3326, 28, 323bitri 205 . . . . . . 7  |-  ( <.
x ,  y >.  e.  ( C  i^i  ( A  X.  B ) )  <-> 
( <. x ,  y
>.  e.  ( C  |`  A )  /\  y  e.  B ) )
3433exbii 1585 . . . . . 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 1882 . . . . . 6  |-  ( E. x ( <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
)  <->  ( E. x <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
) )
3625, 34, 353bitri 205 . . . . 5  |-  ( y  e.  ran  ( C  i^i  ( A  X.  B ) )  <->  ( E. x <. x ,  y
>.  e.  ( C  |`  A )  /\  y  e.  B ) )
3724, 36syl6ibr 161 . . . 4  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  -> 
y  e.  ran  ( C  i^i  ( A  X.  B ) ) ) )
3837ssrdv 3134 . . 3  |-  ( B 
C_  ran  ( C  |`  A )  ->  B  C_ 
ran  ( C  i^i  ( A  X.  B
) ) )
3919, 38impbii 125 . 2  |-  ( B 
C_  ran  ( C  i^i  ( A  X.  B
) )  <->  B  C_  ran  ( C  |`  A ) )
407, 39bitr2i 184 1  |-  ( B 
C_  ran  ( C  |`  A )  <->  ran  ( C  i^i  ( A  X.  B ) )  =  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1335   E.wex 1472    e. wcel 2128   _Vcvv 2712    i^i cin 3101    C_ wss 3102   <.cop 3563    X. cxp 4585   ran crn 4588    |` cres 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-xp 4593  df-rel 4594  df-cnv 4595  df-dm 4597  df-rn 4598  df-res 4599
This theorem is referenced by:  rninxp  5030
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