Theorem rninxp 5074

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 3147	. 2 |- ( B C_ ran ( C |` A ) <-> A. y e. B y e. ran ( C |` A ) )
2		ssrnres 5073	. 2 |- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B )
3		df-ima 4641	. . . . 5 |- ( C " A ) = ran ( C |` A )
4	3	eleq2i 2244	. . . 4 |- ( y e. ( C " A ) <-> y e. ran ( C |` A ) )
5		vex 2742	. . . . 5 |- y e. _V
6	5	elima 4977	. . . 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 2483	. 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 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   i^i cin 3130   C_ wss 3131   class class class wbr 4005   X. cxp 4626   ran crn 4629   |` cres 4630   "cima 4631

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641

This theorem is referenced by:  dminxp  5075  fncnv  5284