MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnxpid Unicode version

Theorem rnxpid 5261
Description: The range of a square cross product. (Contributed by FL, 17-May-2010.)
Assertion
Ref Expression
rnxpid  |-  ran  ( A  X.  A )  =  A

Proof of Theorem rnxpid
StepHypRef Expression
1 rn0 5086 . . 3  |-  ran  (/)  =  (/)
2 xpeq2 4852 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( A  X.  (/) ) )
3 xp0 5250 . . . . 5  |-  ( A  X.  (/) )  =  (/)
42, 3syl6eq 2452 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
54rneqd 5056 . . 3  |-  ( A  =  (/)  ->  ran  ( A  X.  A )  =  ran  (/) )
6 id 20 . . 3  |-  ( A  =  (/)  ->  A  =  (/) )
71, 5, 63eqtr4a 2462 . 2  |-  ( A  =  (/)  ->  ran  ( A  X.  A )  =  A )
8 rnxp 5258 . 2  |-  ( A  =/=  (/)  ->  ran  ( A  X.  A )  =  A )
97, 8pm2.61ine 2643 1  |-  ran  ( A  X.  A )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1649   (/)c0 3588    X. cxp 4835   ran crn 4838
This theorem is referenced by:  sofld  5277  fpwwe2lem13  8473  ustimasn  18211  utopbas  18218  restutop  18220  ovoliunlem1  19351  metideq  24241  mblfinlem  26143
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848
  Copyright terms: Public domain W3C validator