Theorem rnxpm 5095

Description: The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with nonempty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)

Assertion

Ref	Expression
rnxpm	|- ( E. x x e. A -> ran ( A X. B ) = B )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem rnxpm

Step	Hyp	Ref	Expression
1		df-rn 4670	. . 3 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5084	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4863	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4	1, 3	eqtri 2214	. 2 |- ran ( A X. B ) = dom ( B X. A )
5		dmxpm 4882	. 2 |- ( E. x x e. A -> dom ( B X. A ) = B )
6	4, 5	eqtrid 2238	1 |- ( E. x x e. A -> ran ( A X. B ) = B )

Syntax hints:   -> wi 4   = wceq 1364   E.wex 1503   e. wcel 2164   X. cxp 4657   `'ccnv 4658   dom cdm 4659   ran crn 4660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670

This theorem is referenced by:  ssxpbm  5101  ssxp2  5103  xpexr2m  5107  xpima2m  5113  unixpm  5201  djuexb  7103  exmidfodomrlemim  7261  elply2  14881