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Theorem rninxp 4977
Description: Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
rninxp  |-  ( ran  ( C  i^i  ( A  X.  B ) )  =  B  <->  A. y  e.  B  E. x  e.  A  x C
y )
Distinct variable groups:    x, y, A   
y, B    x, C, y
Allowed substitution hint:    B( x)

Proof of Theorem rninxp
StepHypRef Expression
1 dfss3 3082 . 2  |-  ( B 
C_  ran  ( C  |`  A )  <->  A. y  e.  B  y  e.  ran  ( C  |`  A ) )
2 ssrnres 4976 . 2  |-  ( B 
C_  ran  ( C  |`  A )  <->  ran  ( C  i^i  ( A  X.  B ) )  =  B )
3 df-ima 4547 . . . . 5  |-  ( C
" A )  =  ran  ( C  |`  A )
43eleq2i 2204 . . . 4  |-  ( y  e.  ( C " A )  <->  y  e.  ran  ( C  |`  A ) )
5 vex 2684 . . . . 5  |-  y  e. 
_V
65elima 4881 . . . 4  |-  ( y  e.  ( C " A )  <->  E. x  e.  A  x C
y )
74, 6bitr3i 185 . . 3  |-  ( y  e.  ran  ( C  |`  A )  <->  E. x  e.  A  x C
y )
87ralbii 2439 . 2  |-  ( A. y  e.  B  y  e.  ran  ( C  |`  A )  <->  A. y  e.  B  E. x  e.  A  x C
y )
91, 2, 83bitr3i 209 1  |-  ( ran  ( C  i^i  ( A  X.  B ) )  =  B  <->  A. y  e.  B  E. x  e.  A  x C
y )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1331    e. wcel 1480   A.wral 2414   E.wrex 2415    i^i cin 3065    C_ wss 3066   class class class wbr 3924    X. cxp 4532   ran crn 4535    |` cres 4536   "cima 4537
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547
This theorem is referenced by:  dminxp  4978  fncnv  5184
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