Theorem iunconstm 3909

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 3905	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
2		r19.9rmv 3529	. . 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 2187	1 |- ( E. x x e. A -> U_ x e. A B = B )

Syntax hints:   -> wi 4   = wceq 1364   E.wex 1503   e. wcel 2160   E.wrex 2469   U_ciun 3901

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-iun 3903

This theorem is referenced by: (None)