Theorem iunconstm 3952

Description: Indexed union of a constant class, i.e. where B does not depend on x. (Contributed by Jim Kingdon, 15-Aug-2018.)

Assertion

Ref	Expression
iunconstm	|- ( E. x x e. A -> U_ x e. A B = B )

Distinct variable groups:   x, A   x, B

Proof of Theorem iunconstm

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 3948	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
2		r19.9rmv 3563	. . 3 |- ( E. x x e. A -> ( y e. B <-> E. x e. A y e. B ) )
3	1, 2	bitr4id 199	. 2 |- ( E. x x e. A -> ( y e. U_ x e. A B <-> y e. B ) )
4	3	eqrdv 2207	1 |- ( E. x x e. A -> U_ x e. A B = B )

Syntax hints:   -> wi 4   = wceq 1375   E.wex 1518   e. wcel 2180   E.wrex 2489   U_ciun 3944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-iun 3946

This theorem is referenced by: (None)