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Theorem rnxpid 5440
Description: The range of a square Cartesian product. (Contributed by FL, 17-May-2010.)
Assertion
Ref Expression
rnxpid  |-  ran  ( A  X.  A )  =  A

Proof of Theorem rnxpid
StepHypRef Expression
1 rn0 5254 . . 3  |-  ran  (/)  =  (/)
2 xpeq2 5014 . . . . 5  |-  ( A  =  (/)  ->  ( A  X.  A )  =  ( A  X.  (/) ) )
3 xp0 5425 . . . . 5  |-  ( A  X.  (/) )  =  (/)
42, 3syl6eq 2524 . . . 4  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
54rneqd 5230 . . 3  |-  ( A  =  (/)  ->  ran  ( A  X.  A )  =  ran  (/) )
6 id 22 . . 3  |-  ( A  =  (/)  ->  A  =  (/) )
71, 5, 63eqtr4a 2534 . 2  |-  ( A  =  (/)  ->  ran  ( A  X.  A )  =  A )
8 rnxp 5437 . 2  |-  ( A  =/=  (/)  ->  ran  ( A  X.  A )  =  A )
97, 8pm2.61ine 2780 1  |-  ran  ( A  X.  A )  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   (/)c0 3785    X. cxp 4997   ran crn 5000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by:  sofld  5455  fpwwe2lem13  9020  ustimasn  20494  utopbas  20501  restutop  20503  ovoliunlem1  21676  metideq  27536  mblfinlem1  29656
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