Theorem rnxpm 5220

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 4787	. . 3 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5209	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4984	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4	1, 3	eqtri 2259	. 2 |- ran ( A X. B ) = dom ( B X. A )
5		dmxpm 5004	. 2 |- ( E. x x e. A -> dom ( B X. A ) = B )
6	4, 5	eqtrid 2283	1 |- ( E. x x e. A -> ran ( A X. B ) = B )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   E.wex 1545   e. wcel 2209   X. cxp 4774   `'ccnv 4775   dom cdm 4776   ran crn 4777

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-xp 4782  df-rel 4783  df-cnv 4784  df-dm 4786  df-rn 4787

This theorem is used by:  ssxpbm  5226  ssxp2  5228  xpexr2m  5232  xpima2m  5238  unixpm  5326  djuexb  7390  exmidfodomrlemim  7559  elply2  15961