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Theorem rninxp 4707
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  X.  C
Distinct variable groups:   ,,   ,   , C,
Allowed substitution hint:   ()

Proof of Theorem rninxp
StepHypRef Expression
1 dfss3 2929 . 2 
C_  ran  C  |`  ran  C  |`
2 ssrnres 4706 . 2 
C_  ran  C  |`  ran  C  i^i  X.
3 df-ima 4301 . . . . 5  C
" 
ran  C  |`
43eleq2i 2101 . . . 4  C "  ran  C  |`
5 vex 2554 . . . . 5 
_V
65elima 4616 . . . 4  C "  C
74, 6bitr3i 175 . . 3  ran  C  |`  C
87ralbii 2324 . 2  ran  C  |`  C
91, 2, 83bitr3i 199 1  ran  C  i^i  X.  C
Colors of variables: wff set class
Syntax hints:   wb 98   wceq 1242   wcel 1390  wral 2300  wrex 2301    i^i cin 2910    C_ wss 2911   class class class wbr 3755    X. cxp 4286   ran crn 4289    |` cres 4290   "cima 4291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by:  dminxp  4708  fncnv  4908
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