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Theorem dfiun2 4586
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 4584 . 2 (∀𝑥𝐴 𝐵 ∈ V → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
2 dfiun2.1 . . 3 𝐵 ∈ V
32a1i 11 . 2 (𝑥𝐴𝐵 ∈ V)
41, 3mprg 2955 1 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  Vcvv 3231   cuni 4468   ciun 4552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-uni 4469  df-iun 4554
This theorem is referenced by:  fniunfv  6545  funcnvuni  7161  fun11iun  7168  tfrlem8  7525  rdglim2a  7574  rankuni  8764  cardiun  8846  kmlem11  9020  cfslb2n  9128  enfin2i  9181  pwcfsdom  9443  rankcf  9637  tskuni  9643  discmp  21249  cmpsublem  21250  cmpsub  21251
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