Theorem eliin 3741

Description: Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)

Assertion

Ref	Expression
eliin	|- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   C( x)   V( x)

Proof of Theorem eliin

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2151	. . 3 |- ( y = A -> ( y e. C <-> A e. C ) )
2	1	ralbidv 2381	. 2 |- ( y = A -> ( A. x e. B y e. C <-> A. x e. B A e. C ) )
3		df-iin 3739	. 2 |- |^|_ x e. B C = { y | A. x e. B y e. C }
4	2, 3	elab2g 2763	1 |- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1290   e. wcel 1439   A.wral 2360   |^|_ciin 3737

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2622  df-iin 3739

This theorem is referenced by:  iinconstm  3745  iuniin  3746  iinss1  3748  ssiinf  3785  iinss  3787  iinss2  3788  iinab  3797  iundif2ss  3801  iindif2m  3803  iinin2m  3804  elriin  3806  iinpw  3825  xpiindim  4586  cnviinm  4985  iinerm  6378  ixpiinm  6495