Theorem dmxpm 4555

Description: The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
dmxpm	|- ( E. x x e. B -> dom ( A X. B ) = A )

Distinct variable group:   x, B

Allowed substitution hint:   A( x)

Proof of Theorem dmxpm

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2100	. . 3 |- ( x = z -> ( x e. B <-> z e. B ) )
2	1	cbvexv 1795	. 2 |- ( E. x x e. B <-> E. z z e. B )
3		df-xp 4351	. . . 4 |- ( A X. B ) = { <. y , z >. | ( y e. A /\ z e. B ) }
4	3	dmeqi 4536	. . 3 |- dom ( A X. B ) = dom { <. y , z >. | ( y e. A /\ z e. B ) }
5		id 19	. . . . 5 |- ( E. z z e. B -> E. z z e. B )
6	5	ralrimivw 2393	. . . 4 |- ( E. z z e. B -> A. y e. A E. z z e. B )
7		dmopab3 4548	. . . 4 |- ( A. y e. A E. z z e. B <-> dom { <. y , z >. | ( y e. A /\ z e. B ) } = A )
8	6, 7	sylib 127	. . 3 |- ( E. z z e. B -> dom { <. y , z >. | ( y e. A /\ z e. B ) } = A )
9	4, 8	syl5eq 2084	. 2 |- ( E. z z e. B -> dom ( A X. B ) = A )
10	2, 9	sylbi 114	1 |- ( E. x x e. B -> dom ( A X. B ) = A )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   A.wral 2306   {copab 3817   X. cxp 4343   dom cdm 4345

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-dm 4355

This theorem is referenced by:  dmxpinm  4556  xpid11m  4557  rnxpm  4752  ssxpbm  4756  ssxp1  4757  xpexr2m  4762  relrelss  4844  unixpm  4853  xpiderm  6177