Theorem rnxpm 4772

Description: The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with non-empty 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 4374	. . 3 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 4762	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4554	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4	1, 3	eqtri 2101	. 2 |- ran ( A X. B ) = dom ( B X. A )
5		dmxpm 4573	. 2 |- ( E. x x e. A -> dom ( B X. A ) = B )
6	4, 5	syl5eq 2125	1 |- ( E. x x e. A -> ran ( A X. B ) = B )

Syntax hints:   -> wi 4   = wceq 1284   E.wex 1421   e. wcel 1433   X. cxp 4361   `'ccnv 4362   dom cdm 4363   ran crn 4364

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by:  ssxpbm  4776  ssxp2  4778  xpexr2m  4782  xpima2m  4788  unixpm  4873