Theorem iinconstm 3822

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 3453	. . 3 |- ( E. y y e. A -> ( z e. B <-> A. x e. A z e. B ) )
2		vex 2689	. . . 4 |- z e. _V
3		eliin 3818	. . . 4 |- ( z e. _V -> ( z e. |^|_ x e. A B <-> A. x e. A z e. B ) )
4	2, 3	ax-mp 5	. . 3 |- ( z e. |^|_ x e. A B <-> A. x e. A z e. B )
5	1, 4	syl6rbbr 198	. 2 |- ( E. y y e. A -> ( z e. |^|_ x e. A B <-> z e. B ) )
6	5	eqrdv 2137	1 |- ( E. y y e. A -> |^|_ x e. A B = B )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   A.wral 2416   _Vcvv 2686   |^|_ciin 3814

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-iin 3816

This theorem is referenced by:  iin0imm  4092