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Theorem rninxp 5302
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 3330 . 2  |-  ( B 
C_  ran  ( C  |`  A )  <->  A. y  e.  B  y  e.  ran  ( C  |`  A ) )
2 ssrnres 5301 . 2  |-  ( B 
C_  ran  ( C  |`  A )  <->  ran  ( C  i^i  ( A  X.  B ) )  =  B )
3 df-ima 4883 . . . . 5  |-  ( C
" A )  =  ran  ( C  |`  A )
43eleq2i 2499 . . . 4  |-  ( y  e.  ( C " A )  <->  y  e.  ran  ( C  |`  A ) )
5 vex 2951 . . . . 5  |-  y  e. 
_V
65elima 5200 . . . 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 2721 . 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    i^i cin 3311    C_ wss 3312   class class class wbr 4204    X. cxp 4868   ran crn 4871    |` cres 4872   "cima 4873
This theorem is referenced by:  dminxp  5303  fncnv  5507  exfo  5879  brdom3  8398  brdom5  8399  brdom4  8400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883
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