Theorem iunconstm 3857

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 3853	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
2		r19.9rmv 3485	. . 3 |- ( E. x x e. A -> ( y e. B <-> E. x e. A y e. B ) )
3	1, 2	bitr4id 198	. 2 |- ( E. x x e. A -> ( y e. U_ x e. A B <-> y e. B ) )
4	3	eqrdv 2155	1 |- ( E. x x e. A -> U_ x e. A B = B )

Syntax hints:   -> wi 4   = wceq 1335   E.wex 1472   e. wcel 2128   E.wrex 2436   U_ciun 3849

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-iun 3851

This theorem is referenced by: (None)