New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  dfiin2 GIF version

Theorem dfiin2 4002
 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 x A B = {y x A 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 4000 . 2 (x A B V → x A B = {y x A y = B})
2 dfiun2.1 . . 3 B V
32a1i 10 . 2 (x AB V)
41, 3mprg 2683 1 x A B = {y x A y = B}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859  ∩cint 3926  ∩ciin 3970 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-int 3927  df-iin 3972 This theorem is referenced by:  fniinfv  5372
 Copyright terms: Public domain W3C validator