Theorem iunconstm 3979

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 3975	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
2		r19.9rmv 3585	. . 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 2228	1 |- ( E. x x e. A -> U_ x e. A B = B )

Syntax hints:   -> wi 4   = wceq 1397   E.wex 1540   e. wcel 2201   E.wrex 2510   U_ciun 3971

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-iun 3973

This theorem is referenced by: (None)