Theorem rnxpm 5131

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 4704	. . 3 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5120	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4898	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4	1, 3	eqtri 2228	. 2 |- ran ( A X. B ) = dom ( B X. A )
5		dmxpm 4917	. 2 |- ( E. x x e. A -> dom ( B X. A ) = B )
6	4, 5	eqtrid 2252	1 |- ( E. x x e. A -> ran ( A X. B ) = B )

Syntax hints:   -> wi 4   = wceq 1373   E.wex 1516   e. wcel 2178   X. cxp 4691   `'ccnv 4692   dom cdm 4693   ran crn 4694

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by:  ssxpbm  5137  ssxp2  5139  xpexr2m  5143  xpima2m  5149  unixpm  5237  djuexb  7172  exmidfodomrlemim  7340  elply2  15322