Theorem rninxp 5208

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 3229	. 2 |- ( B C_ ran ( C |` A ) <-> A. y e. B y e. ran ( C |` A ) )
2		ssrnres 5207	. 2 |- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B )
3		df-ima 4764	. . . . 5 |- ( C " A ) = ran ( C |` A )
4	3	eleq2i 2301	. . . 4 |- ( y e. ( C " A ) <-> y e. ran ( C |` A ) )
5		vex 2818	. . . . 5 |- y e. _V
6	5	elima 5108	. . . 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 2550	. 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 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   i^i cin 3212   C_ wss 3213   class class class wbr 4111   X. cxp 4749   ran crn 4752   |` cres 4753   "cima 4754

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-cnv 4759  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764

This theorem is referenced by:  dminxp  5209  fncnv  5424