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Theorem ssrnres 5053
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 3348 . . . . 5  |-  ( C  i^i  ( A  X.  B ) )  C_  ( A  X.  B
)
2 rnss 4841 . . . . 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 5042 . . . 4  |-  ran  ( A  X.  B )  C_  B
53, 4sstri 3156 . . 3  |-  ran  ( C  i^i  ( A  X.  B ) )  C_  B
6 eqss 3162 . . 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 935 . 2  |-  ( ran  ( C  i^i  ( A  X.  B ) )  =  B  <->  B  C_  ran  ( C  i^i  ( A  X.  B ) ) )
8 ssid 3167 . . . . . . . 8  |-  A  C_  A
9 ssv 3169 . . . . . . . 8  |-  B  C_  _V
10 xpss12 4718 . . . . . . . 8  |-  ( ( A  C_  A  /\  B  C_  _V )  -> 
( A  X.  B
)  C_  ( A  X.  _V ) )
118, 9, 10mp2an 424 . . . . . . 7  |-  ( A  X.  B )  C_  ( A  X.  _V )
12 sslin 3353 . . . . . . 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 4623 . . . . . 6  |-  ( C  |`  A )  =  ( C  i^i  ( A  X.  _V ) )
1513, 14sseqtrri 3182 . . . . 5  |-  ( C  i^i  ( A  X.  B ) )  C_  ( C  |`  A )
16 rnss 4841 . . . . 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 3155 . . . 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 423 . . 3  |-  ( B 
C_  ran  ( C  i^i  ( A  X.  B
) )  ->  B  C_ 
ran  ( C  |`  A ) )
20 ssel 3141 . . . . . . 7  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  -> 
y  e.  ran  ( C  |`  A ) ) )
21 vex 2733 . . . . . . . 8  |-  y  e. 
_V
2221elrn2 4853 . . . . . . 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 4853 . . . . . 6  |-  ( y  e.  ran  ( C  i^i  ( A  X.  B ) )  <->  E. x <. x ,  y >.  e.  ( C  i^i  ( A  X.  B ) ) )
26 elin 3310 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  ( C  i^i  ( A  X.  B ) )  <-> 
( <. x ,  y
>.  e.  C  /\  <. x ,  y >.  e.  ( A  X.  B ) ) )
27 opelxp 4641 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
2827anbi2i 454 . . . . . . . 8  |-  ( (
<. x ,  y >.  e.  C  /\  <. x ,  y >.  e.  ( A  X.  B ) )  <->  ( <. x ,  y >.  e.  C  /\  ( x  e.  A  /\  y  e.  B
) ) )
2921opelres 4896 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( C  |`  A )  <-> 
( <. x ,  y
>.  e.  C  /\  x  e.  A ) )
3029anbi1i 455 . . . . . . . . 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 1598 . . . . . 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 1895 . . . . . 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 3153 . . 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 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730    i^i cin 3120    C_ wss 3121   <.cop 3586    X. cxp 4609   ran crn 4612    |` cres 4613
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623
This theorem is referenced by:  rninxp  5054
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