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Theorem rninxp 5243
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 3274 . 2  |-  ( B 
C_  ran  ( C  |`  A )  <->  A. y  e.  B  y  e.  ran  ( C  |`  A ) )
2 ssrnres 5242 . 2  |-  ( B 
C_  ran  ( C  |`  A )  <->  ran  ( C  i^i  ( A  X.  B ) )  =  B )
3 df-ima 4824 . . . . 5  |-  ( C
" A )  =  ran  ( C  |`  A )
43eleq2i 2444 . . . 4  |-  ( y  e.  ( C " A )  <->  y  e.  ran  ( C  |`  A ) )
5 vex 2895 . . . . 5  |-  y  e. 
_V
65elima 5141 . . . 4  |-  ( y  e.  ( C " A )  <->  E. x  e.  A  x C
y )
74, 6bitr3i 243 . . 3  |-  ( y  e.  ran  ( C  |`  A )  <->  E. x  e.  A  x C
y )
87ralbii 2666 . 2  |-  ( A. y  e.  B  y  e.  ran  ( C  |`  A )  <->  A. y  e.  B  E. x  e.  A  x C
y )
91, 2, 83bitr3i 267 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 177    = wceq 1649    e. wcel 1717   A.wral 2642   E.wrex 2643    i^i cin 3255    C_ wss 3256   class class class wbr 4146    X. cxp 4809   ran crn 4812    |` cres 4813   "cima 4814
This theorem is referenced by:  dminxp  5244  fncnv  5448  exfo  5819  brdom3  8332  brdom5  8333  brdom4  8334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824
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