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Theorem dfiun2 4689
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 4687 . 2 (∀𝑥𝐴 𝐵 ∈ V → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
2 dfiun2.1 . . 3 𝐵 ∈ V
32a1i 11 . 2 (𝑥𝐴𝐵 ∈ V)
41, 3mprg 3075 1 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  {cab 2757  wrex 3062  Vcvv 3351   cuni 4575   ciun 4655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-uni 4576  df-iun 4657
This theorem is referenced by:  fniunfv  6651  funcnvuni  7270  fun11iun  7277  tfrlem8  7637  rdglim2a  7686  rankuni  8894  cardiun  9012  kmlem11  9188  cfslb2n  9296  enfin2i  9349  pwcfsdom  9611  rankcf  9805  tskuni  9811  discmp  21422  cmpsublem  21423  cmpsub  21424
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