New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  dfiun2 GIF version

Theorem dfiun2 4001
 Description: Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)
Hypothesis
Ref Expression
dfiun2.1 B V
Assertion
Ref Expression
dfiun2 x A B = {y x A y = B}
Distinct variable groups:   x,y   y,A   y,B
Allowed substitution hints:   A(x)   B(x)

Proof of Theorem dfiun2
StepHypRef Expression
1 dfiun2g 3999 . 2 (x A B V → x A B = {y x A y = B})
2 dfiun2.1 . . 3 B V
32a1i 10 . 2 (x AB V)
41, 3mprg 2683 1 x A B = {y x A y = B}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859  ∪cuni 3891  ∪ciun 3969 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-uni 3892  df-iun 3971 This theorem is referenced by:  funcnvuni  5161  fun11iun  5305  fniunfv  5466
 Copyright terms: Public domain W3C validator