Theorem iunconstm 3998

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 3994	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
2		r19.9rmv 3600	. . 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 2230	1 |- ( E. x x e. A -> U_ x e. A B = B )

Syntax hints:   -> wi 4   = wceq 1398   E.wex 1541   e. wcel 2203   E.wrex 2521   U_ciun 3990

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-iun 3992

This theorem is referenced by: (None)