Theorem rnxpm 5157

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 4729	. . 3 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5146	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4923	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4	1, 3	eqtri 2250	. 2 |- ran ( A X. B ) = dom ( B X. A )
5		dmxpm 4943	. 2 |- ( E. x x e. A -> dom ( B X. A ) = B )
6	4, 5	eqtrid 2274	1 |- ( E. x x e. A -> ran ( A X. B ) = B )

Syntax hints:   -> wi 4   = wceq 1395   E.wex 1538   e. wcel 2200   X. cxp 4716   `'ccnv 4717   dom cdm 4718   ran crn 4719

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-dm 4728  df-rn 4729

This theorem is referenced by:  ssxpbm  5163  ssxp2  5165  xpexr2m  5169  xpima2m  5175  unixpm  5263  djuexb  7207  exmidfodomrlemim  7375  elply2  15403