Theorem rninxp 5180

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 3216	. 2 |- ( B C_ ran ( C |` A ) <-> A. y e. B y e. ran ( C |` A ) )
2		ssrnres 5179	. 2 |- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B )
3		df-ima 4738	. . . . 5 |- ( C " A ) = ran ( C |` A )
4	3	eleq2i 2298	. . . 4 |- ( y e. ( C " A ) <-> y e. ran ( C |` A ) )
5		vex 2805	. . . . 5 |- y e. _V
6	5	elima 5081	. . . 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 2538	. 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 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   i^i cin 3199   C_ wss 3200   class class class wbr 4088   X. cxp 4723   ran crn 4726   |` cres 4727   "cima 4728

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:  dminxp  5181  fncnv  5396