Theorem dminxp 4795

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 4555	. . . 4 |- dom ( C i^i ( A X. B ) ) = ran `' ( C i^i ( A X. B ) )
2		cnvin 4761	. . . . . 6 |- `' ( C i^i ( A X. B ) ) = ( `' C i^i `' ( A X. B ) )
3		cnvxp 4772	. . . . . . 7 |- `' ( A X. B ) = ( B X. A )
4	3	ineq2i 3171	. . . . . 6 |- ( `' C i^i `' ( A X. B ) ) = ( `' C i^i ( B X. A ) )
5	2, 4	eqtri 2102	. . . . 5 |- `' ( C i^i ( A X. B ) ) = ( `' C i^i ( B X. A ) )
6	5	rneqi 4590	. . . 4 |- ran `' ( C i^i ( A X. B ) ) = ran ( `' C i^i ( B X. A ) )
7	1, 6	eqtri 2102	. . 3 |- dom ( C i^i ( A X. B ) ) = ran ( `' C i^i ( B X. A ) )
8	7	eqeq1i 2089	. 2 |- ( dom ( C i^i ( A X. B ) ) = A <-> ran ( `' C i^i ( B X. A ) ) = A )
9		rninxp 4794	. 2 |- ( ran ( `' C i^i ( B X. A ) ) = A <-> A. x e. A E. y e. B y `' C x )
10		vex 2605	. . . . 5 |- y e. _V
11		vex 2605	. . . . 5 |- x e. _V
12	10, 11	brcnv 4546	. . . 4 |- ( y `' C x <-> x C y )
13	12	rexbii 2374	. . 3 |- ( E. y e. B y `' C x <-> E. y e. B x C y )
14	13	ralbii 2373	. 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 204	1 |- ( dom ( C i^i ( A X. B ) ) = A <-> A. x e. A E. y e. B x C y )

Syntax hints:   <-> wb 103   = wceq 1285   A.wral 2349   E.wrex 2350   i^i cin 2973   class class class wbr 3793   X. cxp 4369   `'ccnv 4370   dom cdm 4371   ran crn 4372

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384

This theorem is referenced by: (None)