Theorem dminxp 5574

Description: Domain of the intersection with a Cartesian 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 5316	. . . 4 |- dom ( C i^i ( A X. B ) ) = ran `' ( C i^i ( A X. B ) )
2		cnvin 5540	. . . . . 6 |- `' ( C i^i ( A X. B ) ) = ( `' C i^i `' ( A X. B ) )
3		cnvxp 5551	. . . . . . 7 |- `' ( A X. B ) = ( B X. A )
4	3	ineq2i 3811	. . . . . 6 |- ( `' C i^i `' ( A X. B ) ) = ( `' C i^i ( B X. A ) )
5	2, 4	eqtri 2644	. . . . 5 |- `' ( C i^i ( A X. B ) ) = ( `' C i^i ( B X. A ) )
6	5	rneqi 5352	. . . 4 |- ran `' ( C i^i ( A X. B ) ) = ran ( `' C i^i ( B X. A ) )
7	1, 6	eqtri 2644	. . 3 |- dom ( C i^i ( A X. B ) ) = ran ( `' C i^i ( B X. A ) )
8	7	eqeq1i 2627	. 2 |- ( dom ( C i^i ( A X. B ) ) = A <-> ran ( `' C i^i ( B X. A ) ) = A )
9		rninxp 5573	. 2 |- ( ran ( `' C i^i ( B X. A ) ) = A <-> A. x e. A E. y e. B y `' C x )
10		vex 3203	. . . . 5 |- y e. _V
11		vex 3203	. . . . 5 |- x e. _V
12	10, 11	brcnv 5305	. . . 4 |- ( y `' C x <-> x C y )
13	12	rexbii 3041	. . 3 |- ( E. y e. B y `' C x <-> E. y e. B x C y )
14	13	ralbii 2980	. 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 286	1 |- ( dom ( C i^i ( A X. B ) ) = A <-> A. x e. A E. y e. B x C y )

Syntax hints:   <-> wb 196   = wceq 1483   A.wral 2912   E.wrex 2913   i^i cin 3573   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  trust  22033