Theorem rninxp 5573

Description: Range of the intersection with a Cartesian 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 3592	. 2 |- ( B C_ ran ( C |` A ) <-> A. y e. B y e. ran ( C |` A ) )
2		ssrnres 5572	. 2 |- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B )
3		df-ima 5127	. . . . 5 |- ( C " A ) = ran ( C |` A )
4	3	eleq2i 2693	. . . 4 |- ( y e. ( C " A ) <-> y e. ran ( C |` A ) )
5		vex 3203	. . . . 5 |- y e. _V
6	5	elima 5471	. . . 4 |- ( y e. ( C " A ) <-> E. x e. A x C y )
7	4, 6	bitr3i 266	. . 3 |- ( y e. ran ( C |` A ) <-> E. x e. A x C y )
8	7	ralbii 2980	. 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 290	1 |- ( ran ( C i^i ( A X. B ) ) = B <-> A. y e. B E. x e. A x C y )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   class class class wbr 4653   X. cxp 5112   ran crn 5115   |` cres 5116   "cima 5117

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:  dminxp  5574  fncnv  5962  exfo  6377  brdom3  9350  brdom5  9351  brdom4  9352