Theorem iinconstm 3687

Description: Indexed intersection of a constant class, i.e. where B does not depend on x. (Contributed by Jim Kingdon, 19-Dec-2018.)

Assertion

Ref	Expression
iinconstm	|- ( E. y y e. A -> |^|_ x e. A B = B )

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

Allowed substitution hint:   B( y)

Proof of Theorem iinconstm

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.3rmv 3332	. . 3 |- ( E. y y e. A -> ( z e. B <-> A. x e. A z e. B ) )
2		vex 2604	. . . 4 |- z e. _V
3		eliin 3683	. . . 4 |- ( z e. _V -> ( z e. |^|_ x e. A B <-> A. x e. A z e. B ) )
4	2, 3	ax-mp 7	. . 3 |- ( z e. |^|_ x e. A B <-> A. x e. A z e. B )
5	1, 4	syl6rbbr 197	. 2 |- ( E. y y e. A -> ( z e. |^|_ x e. A B <-> z e. B ) )
6	5	eqrdv 2079	1 |- ( E. y y e. A -> |^|_ x e. A B = B )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   A.wral 2348   _Vcvv 2601   |^|_ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-iin 3681

This theorem is referenced by:  iin0imm  3942