MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfiin2 Structured version   Visualization version   GIF version

Theorem dfiin2 5033
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 5031 . 2 (∀𝑥𝐴 𝐵 ∈ V → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
2 dfiun2.1 . . 3 𝐵 ∈ V
32a1i 11 . 2 (𝑥𝐴𝐵 ∈ V)
41, 3mprg 3066 1 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  {cab 2713  wrex 3069  Vcvv 3479   cint 4945   ciin 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-v 3481  df-int 4946  df-iin 4993
This theorem is referenced by:  fniinfv  6986  scott0  9927  cfval2  10301  cflim3  10303  cflim2  10304  cfss  10306  hauscmplem  23415  ptbasfi  23590  dihglblem5  41301  dihglb2  41345  intima0  43666
  Copyright terms: Public domain W3C validator