Theorem rninxp 5178

Description: Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

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

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

Allowed substitution hint:   B( x)

Proof of Theorem rninxp

Step	Hyp	Ref	Expression
1		dfss3 3214	. 2 |- ( B C_ ran ( C |` A ) <-> A. y e. B y e. ran ( C |` A ) )
2		ssrnres 5177	. 2 |- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B )
3		df-ima 4736	. . . . 5 |- ( C " A ) = ran ( C |` A )
4	3	eleq2i 2296	. . . 4 |- ( y e. ( C " A ) <-> y e. ran ( C |` A ) )
5		vex 2803	. . . . 5 |- y e. _V
6	5	elima 5079	. . . 4 |- ( y e. ( C " A ) <-> E. x e. A x C y )
7	4, 6	bitr3i 186	. . 3 |- ( y e. ran ( C |` A ) <-> E. x e. A x C y )
8	7	ralbii 2536	. 2 |- ( A. y e. B y e. ran ( C |` A ) <-> A. y e. B E. x e. A x C y )
9	1, 2, 8	3bitr3i 210	1 |- ( ran ( C i^i ( A X. B ) ) = B <-> A. y e. B E. x e. A x C y )

Syntax hints:   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   i^i cin 3197   C_ wss 3198   class class class wbr 4086   X. cxp 4721   ran crn 4724   |` cres 4725   "cima 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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736

This theorem is referenced by:  dminxp  5179  fncnv  5393