Theorem dfiun2 2583

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 2582	. 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 1698	1 |- U_x e. A B = U.{y | E.x e. A y = B}

Syntax hints:   = wceq 955   e. wcel 957  {cab 1462  E.wrex 1644  Vcvv 1808  U.cuni 2499  U_ciun 2562

This theorem is referenced by:  funcnvuni 3560  fniunfv 3860  iunon 3904  tfrlem8 3913  rdglim2a 3945  rankuni 4681  kmlem11 4758  cardiun 4842

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ral 1647  df-rex 1648  df-v 1809  df-uni 2500  df-iun 2564

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