Theorem iinconstm 3747

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 3378	. . 3 |- ( E. y y e. A -> ( z e. B <-> A. x e. A z e. B ) )
2		vex 2625	. . . 4 |- z e. _V
3		eliin 3743	. . . 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 198	. 2 |- ( E. y y e. A -> ( z e. |^|_ x e. A B <-> z e. B ) )
6	5	eqrdv 2087	1 |- ( E. y y e. A -> |^|_ x e. A B = B )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1290   E.wex 1427   e. wcel 1439   A.wral 2360   _Vcvv 2622   |^|_ciin 3739

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 2624  df-iin 3741

This theorem is referenced by:  iin0imm  4011