Theorem rnxpm 4860

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 4449	. . 3 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 4850	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4637	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4	1, 3	eqtri 2108	. 2 |- ran ( A X. B ) = dom ( B X. A )
5		dmxpm 4656	. 2 |- ( E. x x e. A -> dom ( B X. A ) = B )
6	4, 5	syl5eq 2132	1 |- ( E. x x e. A -> ran ( A X. B ) = B )

Syntax hints:   -> wi 4   = wceq 1289   E.wex 1426   e. wcel 1438   X. cxp 4436   `'ccnv 4437   dom cdm 4438   ran crn 4439

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449

This theorem is referenced by:  ssxpbm  4866  ssxp2  4868  xpexr2m  4872  xpima2m  4878  unixpm  4966  exmidfodomrlemim  6827