Theorem dfiun2 2587

Description: Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44.

Hypothesis

Ref	Expression
dfiun2.1	|- B e. V

Assertion

Ref	Expression
dfiun2	|- U_x e. A B = U.{y | E.x e. A y = B}

Distinct variable groups:   x,y,A   y,B

Proof of Theorem dfiun2

Step	Hyp	Ref	Expression
1		dfiun2g 2586	. 2 |- (A.x e. A B e. V -> U_x e. A B = U.{y | E.x e. A y = B})
2		dfiun2.1	. . 3 |- B e. V
3	2	a1i 8	. 2 |- (x e. A -> B e. V)
4	1, 3	mprg 1700	1 |- U_x e. A B = U.{y | E.x e. A y = B}

Syntax hints:   = wceq 956   e. wcel 958  {cab 1463  E.wrex 1646  Vcvv 1811  U.cuni 2503  U_ciun 2566

This theorem is referenced by:  funcnvuni 3564  fniunfv 3865  iunon 3909  tfrlem8 3918  rdglim2a 3950  rankuni 4698  kmlem11 4775  cardiun 4859

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-uni 2504  df-iun 2568

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