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Theorem ssrnres 5262
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 3659 . . . . 5  |-  ( C  i^i  ( A  X.  B ) )  C_  ( A  X.  B
)
2 rnss 5051 . . . . 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 5256 . . . 4  |-  ran  ( A  X.  B )  C_  B
53, 4sstri 3450 . . 3  |-  ran  ( C  i^i  ( A  X.  B ) )  C_  B
6 eqss 3456 . . 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 919 . 2  |-  ( ran  ( C  i^i  ( A  X.  B ) )  =  B  <->  B  C_  ran  ( C  i^i  ( A  X.  B ) ) )
8 ssid 3460 . . . . . . . 8  |-  A  C_  A
9 ssv 3461 . . . . . . . 8  |-  B  C_  _V
10 xpss12 4928 . . . . . . . 8  |-  ( ( A  C_  A  /\  B  C_  _V )  -> 
( A  X.  B
)  C_  ( A  X.  _V ) )
118, 9, 10mp2an 670 . . . . . . 7  |-  ( A  X.  B )  C_  ( A  X.  _V )
12 sslin 3664 . . . . . . 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 4834 . . . . . 6  |-  ( C  |`  A )  =  ( C  i^i  ( A  X.  _V ) )
1513, 14sseqtr4i 3474 . . . . 5  |-  ( C  i^i  ( A  X.  B ) )  C_  ( C  |`  A )
16 rnss 5051 . . . . 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 3449 . . . 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 669 . . 3  |-  ( B 
C_  ran  ( C  i^i  ( A  X.  B
) )  ->  B  C_ 
ran  ( C  |`  A ) )
20 ssel 3435 . . . . . . 7  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  -> 
y  e.  ran  ( C  |`  A ) ) )
21 vex 3061 . . . . . . . 8  |-  y  e. 
_V
2221elrn2 5062 . . . . . . 7  |-  ( y  e.  ran  ( C  |`  A )  <->  E. x <. x ,  y >.  e.  ( C  |`  A ) )
2320, 22syl6ib 226 . . . . . 6  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  ->  E. x <. x ,  y
>.  e.  ( C  |`  A ) ) )
2423ancrd 552 . . . . 5  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  -> 
( E. x <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
) ) )
2521elrn2 5062 . . . . . 6  |-  ( y  e.  ran  ( C  i^i  ( A  X.  B ) )  <->  E. x <. x ,  y >.  e.  ( C  i^i  ( A  X.  B ) ) )
26 elin 3625 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  ( C  i^i  ( A  X.  B ) )  <-> 
( <. x ,  y
>.  e.  C  /\  <. x ,  y >.  e.  ( A  X.  B ) ) )
27 opelxp 4852 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
2827anbi2i 692 . . . . . . . 8  |-  ( (
<. x ,  y >.  e.  C  /\  <. x ,  y >.  e.  ( A  X.  B ) )  <->  ( <. x ,  y >.  e.  C  /\  ( x  e.  A  /\  y  e.  B
) ) )
2921opelres 5098 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( C  |`  A )  <-> 
( <. x ,  y
>.  e.  C  /\  x  e.  A ) )
3029anbi1i 693 . . . . . . . . 9  |-  ( (
<. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
)  <->  ( ( <.
x ,  y >.  e.  C  /\  x  e.  A )  /\  y  e.  B ) )
31 anass 647 . . . . . . . . 9  |-  ( ( ( <. x ,  y
>.  e.  C  /\  x  e.  A )  /\  y  e.  B )  <->  ( <. x ,  y >.  e.  C  /\  ( x  e.  A  /\  y  e.  B
) ) )
3230, 31bitr2i 250 . . . . . . . 8  |-  ( (
<. x ,  y >.  e.  C  /\  (
x  e.  A  /\  y  e.  B )
)  <->  ( <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
) )
3326, 28, 323bitri 271 . . . . . . 7  |-  ( <.
x ,  y >.  e.  ( C  i^i  ( A  X.  B ) )  <-> 
( <. x ,  y
>.  e.  ( C  |`  A )  /\  y  e.  B ) )
3433exbii 1688 . . . . . 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 1795 . . . . . 6  |-  ( E. x ( <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
)  <->  ( E. x <. x ,  y >.  e.  ( C  |`  A )  /\  y  e.  B
) )
3625, 34, 353bitri 271 . . . . 5  |-  ( y  e.  ran  ( C  i^i  ( A  X.  B ) )  <->  ( E. x <. x ,  y
>.  e.  ( C  |`  A )  /\  y  e.  B ) )
3724, 36syl6ibr 227 . . . 4  |-  ( B 
C_  ran  ( C  |`  A )  ->  (
y  e.  B  -> 
y  e.  ran  ( C  i^i  ( A  X.  B ) ) ) )
3837ssrdv 3447 . . 3  |-  ( B 
C_  ran  ( C  |`  A )  ->  B  C_ 
ran  ( C  i^i  ( A  X.  B
) ) )
3919, 38impbii 188 . 2  |-  ( B 
C_  ran  ( C  i^i  ( A  X.  B
) )  <->  B  C_  ran  ( C  |`  A ) )
407, 39bitr2i 250 1  |-  ( B 
C_  ran  ( C  |`  A )  <->  ran  ( C  i^i  ( A  X.  B ) )  =  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   _Vcvv 3058    i^i cin 3412    C_ wss 3413   <.cop 3977    X. cxp 4820   ran crn 4823    |` cres 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-rel 4829  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834
This theorem is referenced by:  rninxp  5263
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