Theorem rninxp 4355

Description: Range of the intersection with a cross product. (The proof was 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 xCy)

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

Proof of Theorem rninxp

Step	Hyp	Ref	Expression
1		dfss3 2611	. 2 |- (B C_ ran ( C |` A) <-> A.y e. B y e. ran ( C |` A))
2		ssrnres 4354	. 2 |- (B C_ ran ( C |` A) <-> ran ( C i^i (A X. B)) = B)
3		df-ima 4007	. . . . 5 |- (C"A) = ran ( C |` A)
4	3	eleq2i 1961	. . . 4 |- (y e. (C"A) <-> y e. ran ( C |` A))
5		visset 2295	. . . . 5 |- y e. _V
6	5	elima 4270	. . . 4 |- (y e. (C"A) <-> E.x e. A xCy)
7	4, 6	bitr3i 192	. . 3 |- (y e. ran ( C |` A) <-> E.x e. A xCy)
8	7	ralbii 2127	. 2 |- (A.y e. B y e. ran ( C |` A) <-> A.y e. B E.x e. A xCy)
9	1, 2, 8	3bitr3i 198	1 |- (ran ( C i^i (A X. B)) = B <-> A.y e. B E.x e. A xCy)

Syntax hints:   <-> wb 163   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   i^i cin 2592   C_ wss 2593   class class class wbr 3338   X. cxp 3984  ran crn 3987   |` cres 3988  "cima 3989

This theorem is referenced by:  dminxp 4357  fncnv 4479  exfo 4795  brdom3 5963  brdom5 5964  brdom4 5965

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007

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