Theorem dminxp 5181

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 4923	. . . 4 |- dom ( C i^i ( A X. B ) ) = ran `' ( C i^i ( A X. B ) )
2		cnvin 5144	. . . . . 6 |- `' ( C i^i ( A X. B ) ) = ( `' C i^i `' ( A X. B ) )
3		cnvxp 5155	. . . . . . 7 |- `' ( A X. B ) = ( B X. A )
4	3	ineq2i 3405	. . . . . 6 |- ( `' C i^i `' ( A X. B ) ) = ( `' C i^i ( B X. A ) )
5	2, 4	eqtri 2252	. . . . 5 |- `' ( C i^i ( A X. B ) ) = ( `' C i^i ( B X. A ) )
6	5	rneqi 4960	. . . 4 |- ran `' ( C i^i ( A X. B ) ) = ran ( `' C i^i ( B X. A ) )
7	1, 6	eqtri 2252	. . 3 |- dom ( C i^i ( A X. B ) ) = ran ( `' C i^i ( B X. A ) )
8	7	eqeq1i 2239	. 2 |- ( dom ( C i^i ( A X. B ) ) = A <-> ran ( `' C i^i ( B X. A ) ) = A )
9		rninxp 5180	. 2 |- ( ran ( `' C i^i ( B X. A ) ) = A <-> A. x e. A E. y e. B y `' C x )
10		vex 2805	. . . . 5 |- y e. _V
11		vex 2805	. . . . 5 |- x e. _V
12	10, 11	brcnv 4913	. . . 4 |- ( y `' C x <-> x C y )
13	12	rexbii 2539	. . 3 |- ( E. y e. B y `' C x <-> E. y e. B x C y )
14	13	ralbii 2538	. 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 206	1 |- ( dom ( C i^i ( A X. B ) ) = A <-> A. x e. A E. y e. B x C y )

Syntax hints:   <-> wb 105   = wceq 1397   A.wral 2510   E.wrex 2511   i^i cin 3199   class class class wbr 4088   X. cxp 4723   `'ccnv 4724   dom cdm 4725   ran crn 4726

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by: (None)