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Theorem rninxp 5117
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 3170 . 2  |-  ( B 
C_  ran  ( C  |`  A )  <->  A. y  e.  B  y  e.  ran  ( C  |`  A ) )
2 ssrnres 5116 . 2  |-  ( B 
C_  ran  ( C  |`  A )  <->  ran  ( C  i^i  ( A  X.  B ) )  =  B )
3 df-ima 4702 . . . . 5  |-  ( C
" A )  =  ran  ( C  |`  A )
43eleq2i 2347 . . . 4  |-  ( y  e.  ( C " A )  <->  y  e.  ran  ( C  |`  A ) )
5 vex 2791 . . . . 5  |-  y  e. 
_V
65elima 5017 . . . 4  |-  ( y  e.  ( C " A )  <->  E. x  e.  A  x C
y )
74, 6bitr3i 242 . . 3  |-  ( y  e.  ran  ( C  |`  A )  <->  E. x  e.  A  x C
y )
87ralbii 2567 . 2  |-  ( A. y  e.  B  y  e.  ran  ( C  |`  A )  <->  A. y  e.  B  E. x  e.  A  x C
y )
91, 2, 83bitr3i 266 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 176    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   class class class wbr 4023    X. cxp 4687   ran crn 4690    |` cres 4691   "cima 4692
This theorem is referenced by:  dminxp  5118  fncnv  5314  exfo  5678  brdom3  8153  brdom5  8154  brdom4  8155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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