Theorem iinconstm 3882

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		vex 2733	. . . 4 |- z e. _V
2		eliin 3878	. . . 4 |- ( z e. _V -> ( z e. |^|_ x e. A B <-> A. x e. A z e. B ) )
3	1, 2	ax-mp 5	. . 3 |- ( z e. |^|_ x e. A B <-> A. x e. A z e. B )
4		r19.3rmv 3505	. . 3 |- ( E. y y e. A -> ( z e. B <-> A. x e. A z e. B ) )
5	3, 4	bitr4id 198	. 2 |- ( E. y y e. A -> ( z e. |^|_ x e. A B <-> z e. B ) )
6	5	eqrdv 2168	1 |- ( E. y y e. A -> |^|_ x e. A B = B )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   A.wral 2448   _Vcvv 2730   |^|_ciin 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-iin 3876

This theorem is referenced by:  iin0imm  4154