Theorem dfiun3 5021

Description: Alternate definition of indexed union when B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
dfiun3.1	|- B e. _V

Assertion

Ref	Expression
dfiun3	|- U_ x e. A B = U. ran ( x e. A |-> B )

Proof of Theorem dfiun3

Step	Hyp	Ref	Expression
1		dfiun3g 5019	. 2 |- ( A. x e. A B e. _V -> U_ x e. A B = U. ran ( x e. A |-> B ) )
2		dfiun3.1	. . 3 |- B e. _V
3	2	a1i 9	. 2 |- ( x e. A -> B e. _V )
4	1, 3	mprg 2601	1 |- U_ x e. A B = U. ran ( x e. A |-> B )

Syntax hints:   = wceq 1398   e. wcel 2205   _Vcvv 2815   U.cuni 3919   U_ciun 3996   |-> cmpt 4176   ran crn 4755

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-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-cnv 4762  df-dm 4764  df-rn 4765

This theorem is referenced by:  tgrest  15160