Theorem rnxpm 5033

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 4615	. . 3 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5022	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4805	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4	1, 3	eqtri 2186	. 2 |- ran ( A X. B ) = dom ( B X. A )
5		dmxpm 4824	. 2 |- ( E. x x e. A -> dom ( B X. A ) = B )
6	4, 5	syl5eq 2211	1 |- ( E. x x e. A -> ran ( A X. B ) = B )

Syntax hints:   -> wi 4   = wceq 1343   E.wex 1480   e. wcel 2136   X. cxp 4602   `'ccnv 4603   dom cdm 4604   ran crn 4605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by:  ssxpbm  5039  ssxp2  5041  xpexr2m  5045  xpima2m  5051  unixpm  5139  djuexb  7009  exmidfodomrlemim  7157