Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfiun2 Structured version   Visualization version   GIF version

Theorem dfiun2 4955
 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 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 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 4952 . 2 (∀𝑥𝐴 𝐵 ∈ V → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
2 dfiun2.1 . . 3 𝐵 ∈ V
32a1i 11 . 2 (𝑥𝐴𝐵 ∈ V)
41, 3mprg 3157 1 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1530   ∈ wcel 2107  {cab 2804  ∃wrex 3144  Vcvv 3500  ∪ cuni 4837  ∪ ciun 4917 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-v 3502  df-uni 4838  df-iun 4919 This theorem is referenced by:  fniunfv  7000  funcnvuni  7624  fiunw  7632  f1iunw  7633  fiun  7635  f1iun  7636  tfrlem8  8011  rdglim2a  8060  rankuni  9281  cardiun  9400  kmlem11  9575  cfslb2n  9679  enfin2i  9732  pwcfsdom  9994  rankcf  10188  tskuni  10194  discmp  21925  cmpsublem  21926  cmpsub  21927
 Copyright terms: Public domain W3C validator