Theorem rnxpm 5012

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 4594	. . 3 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5001	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4784	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4	1, 3	eqtri 2178	. 2 |- ran ( A X. B ) = dom ( B X. A )
5		dmxpm 4803	. 2 |- ( E. x x e. A -> dom ( B X. A ) = B )
6	4, 5	syl5eq 2202	1 |- ( E. x x e. A -> ran ( A X. B ) = B )

Syntax hints:   -> wi 4   = wceq 1335   E.wex 1472   e. wcel 2128   X. cxp 4581   `'ccnv 4582   dom cdm 4583   ran crn 4584

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4589  df-rel 4590  df-cnv 4591  df-dm 4593  df-rn 4594

This theorem is referenced by:  ssxpbm  5018  ssxp2  5020  xpexr2m  5024  xpima2m  5030  unixpm  5118  djuexb  6978  exmidfodomrlemim  7119