Theorem dminxp 5127

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 4870	. . . 4 |- dom ( C i^i ( A X. B ) ) = ran `' ( C i^i ( A X. B ) )
2		cnvin 5090	. . . . . 6 |- `' ( C i^i ( A X. B ) ) = ( `' C i^i `' ( A X. B ) )
3		cnvxp 5101	. . . . . . 7 |- `' ( A X. B ) = ( B X. A )
4	3	ineq2i 3371	. . . . . 6 |- ( `' C i^i `' ( A X. B ) ) = ( `' C i^i ( B X. A ) )
5	2, 4	eqtri 2226	. . . . 5 |- `' ( C i^i ( A X. B ) ) = ( `' C i^i ( B X. A ) )
6	5	rneqi 4906	. . . 4 |- ran `' ( C i^i ( A X. B ) ) = ran ( `' C i^i ( B X. A ) )
7	1, 6	eqtri 2226	. . 3 |- dom ( C i^i ( A X. B ) ) = ran ( `' C i^i ( B X. A ) )
8	7	eqeq1i 2213	. 2 |- ( dom ( C i^i ( A X. B ) ) = A <-> ran ( `' C i^i ( B X. A ) ) = A )
9		rninxp 5126	. 2 |- ( ran ( `' C i^i ( B X. A ) ) = A <-> A. x e. A E. y e. B y `' C x )
10		vex 2775	. . . . 5 |- y e. _V
11		vex 2775	. . . . 5 |- x e. _V
12	10, 11	brcnv 4861	. . . 4 |- ( y `' C x <-> x C y )
13	12	rexbii 2513	. . 3 |- ( E. y e. B y `' C x <-> E. y e. B x C y )
14	13	ralbii 2512	. 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 1373   A.wral 2484   E.wrex 2485   i^i cin 3165   class class class wbr 4044   X. cxp 4673   `'ccnv 4674   dom cdm 4675   ran crn 4676

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688

This theorem is referenced by: (None)