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Theorem dfiin2 5028
Description: Alternate definition of indexed intersection when 𝐵 is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Hypothesis
Ref Expression
dfiun2.1 𝐵 ∈ V
Assertion
Ref Expression
dfiin2 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 dfiin2
StepHypRef Expression
1 dfiin2g 5026 . 2 (∀𝑥𝐴 𝐵 ∈ V → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
2 dfiun2.1 . . 3 𝐵 ∈ V
32a1i 11 . 2 (𝑥𝐴𝐵 ∈ V)
41, 3mprg 3059 1 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  {cab 2701  wrex 3062  Vcvv 3466   cint 4941   ciin 4989
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-v 3468  df-int 4942  df-iin 4991
This theorem is referenced by:  fniinfv  6960  scott0  9878  cfval2  10252  cflim3  10254  cflim2  10255  cfss  10257  hauscmplem  23254  ptbasfi  23429  dihglblem5  40673  dihglb2  40717  intima0  42949
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