Theorem rnxpm 5197

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 4765	. . 3 |- ran ( A X. B ) = dom `' ( A X. B )
2		cnvxp 5186	. . . 4 |- `' ( A X. B ) = ( B X. A )
3	2	dmeqi 4962	. . 3 |- dom `' ( A X. B ) = dom ( B X. A )
4	1, 3	eqtri 2255	. 2 |- ran ( A X. B ) = dom ( B X. A )
5		dmxpm 4982	. 2 |- ( E. x x e. A -> dom ( B X. A ) = B )
6	4, 5	eqtrid 2279	1 |- ( E. x x e. A -> ran ( A X. B ) = B )

Syntax hints:   -> wi 4   = wceq 1398   E.wex 1541   e. wcel 2205   X. cxp 4752   `'ccnv 4753   dom cdm 4754   ran crn 4755

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762  df-dm 4764  df-rn 4765

This theorem is referenced by:  ssxpbm  5203  ssxp2  5205  xpexr2m  5209  xpima2m  5215  unixpm  5303  djuexb  7348  exmidfodomrlemim  7517  elply2  15726