Theorem iinconstm 3830

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 2692	. . . 4 |- z e. _V
2		eliin 3826	. . . 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 3458	. . 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 2138	1 |- ( E. y y e. A -> |^|_ x e. A B = B )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   A.wral 2417   _Vcvv 2689   |^|_ciin 3822

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-iin 3824

This theorem is referenced by:  iin0imm  4100