Theorem dminxp 4991

Description: Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)

Assertion

Ref	Expression
dminxp	|- ( dom ( C i^i ( A X. B ) ) = A <-> A. x e. A E. y e. B x C y )

Distinct variable groups:   x, A   x, y, B   x, C, y

Allowed substitution hint:   A( y)

Proof of Theorem dminxp

Step	Hyp	Ref	Expression
1		dfdm4 4739	. . . 4 |- dom ( C i^i ( A X. B ) ) = ran `' ( C i^i ( A X. B ) )
2		cnvin 4954	. . . . . 6 |- `' ( C i^i ( A X. B ) ) = ( `' C i^i `' ( A X. B ) )
3		cnvxp 4965	. . . . . . 7 |- `' ( A X. B ) = ( B X. A )
4	3	ineq2i 3279	. . . . . 6 |- ( `' C i^i `' ( A X. B ) ) = ( `' C i^i ( B X. A ) )
5	2, 4	eqtri 2161	. . . . 5 |- `' ( C i^i ( A X. B ) ) = ( `' C i^i ( B X. A ) )
6	5	rneqi 4775	. . . 4 |- ran `' ( C i^i ( A X. B ) ) = ran ( `' C i^i ( B X. A ) )
7	1, 6	eqtri 2161	. . 3 |- dom ( C i^i ( A X. B ) ) = ran ( `' C i^i ( B X. A ) )
8	7	eqeq1i 2148	. 2 |- ( dom ( C i^i ( A X. B ) ) = A <-> ran ( `' C i^i ( B X. A ) ) = A )
9		rninxp 4990	. 2 |- ( ran ( `' C i^i ( B X. A ) ) = A <-> A. x e. A E. y e. B y `' C x )
10		vex 2692	. . . . 5 |- y e. _V
11		vex 2692	. . . . 5 |- x e. _V
12	10, 11	brcnv 4730	. . . 4 |- ( y `' C x <-> x C y )
13	12	rexbii 2445	. . 3 |- ( E. y e. B y `' C x <-> E. y e. B x C y )
14	13	ralbii 2444	. 2 |- ( A. x e. A E. y e. B y `' C x <-> A. x e. A E. y e. B x C y )
15	8, 9, 14	3bitri 205	1 |- ( dom ( C i^i ( A X. B ) ) = A <-> A. x e. A E. y e. B x C y )

Syntax hints:   <-> wb 104   = wceq 1332   A.wral 2417   E.wrex 2418   i^i cin 3075   class class class wbr 3937   X. cxp 4545   `'ccnv 4546   dom cdm 4547   ran crn 4548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560

This theorem is referenced by: (None)