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Theorem rninxp 4874
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 3015 . 2  |-  ( B 
C_  ran  ( C  |`  A )  <->  A. y  e.  B  y  e.  ran  ( C  |`  A ) )
2 ssrnres 4873 . 2  |-  ( B 
C_  ran  ( C  |`  A )  <->  ran  ( C  i^i  ( A  X.  B ) )  =  B )
3 df-ima 4451 . . . . 5  |-  ( C
" A )  =  ran  ( C  |`  A )
43eleq2i 2154 . . . 4  |-  ( y  e.  ( C " A )  <->  y  e.  ran  ( C  |`  A ) )
5 vex 2622 . . . . 5  |-  y  e. 
_V
65elima 4779 . . . 4  |-  ( y  e.  ( C " A )  <->  E. x  e.  A  x C
y )
74, 6bitr3i 184 . . 3  |-  ( y  e.  ran  ( C  |`  A )  <->  E. x  e.  A  x C
y )
87ralbii 2384 . 2  |-  ( A. y  e.  B  y  e.  ran  ( C  |`  A )  <->  A. y  e.  B  E. x  e.  A  x C
y )
91, 2, 83bitr3i 208 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 103    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360    i^i cin 2998    C_ wss 2999   class class class wbr 3845    X. cxp 4436   ran crn 4439    |` cres 4440   "cima 4441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451
This theorem is referenced by:  dminxp  4875  fncnv  5080
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