Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iunin1f Structured version   Unicode version

Theorem iunin1f 28173
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4352 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) (Revised by Thierry Arnoux, 2-May-2020.)
Hypothesis
Ref Expression
iunin1f.1  |-  F/_ x C
Assertion
Ref Expression
iunin1f  |-  U_ x  e.  A  ( B  i^i  C )  =  (
U_ x  e.  A  B  i^i  C )

Proof of Theorem iunin1f
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfcv 2580 . . . . . 6  |-  F/_ x
y
2 iunin1f.1 . . . . . 6  |-  F/_ x C
31, 2nfel 2593 . . . . 5  |-  F/ x  y  e.  C
43r19.41 2978 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  y  e.  C )  <->  ( E. x  e.  A  y  e.  B  /\  y  e.  C )
)
5 elin 3649 . . . . 5  |-  ( y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e.  C ) )
65rexbii 2924 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  i^i  C )  <->  E. x  e.  A  ( y  e.  B  /\  y  e.  C
) )
7 eliun 4304 . . . . 5  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
87anbi1i 699 . . . 4  |-  ( ( y  e.  U_ x  e.  A  B  /\  y  e.  C )  <->  ( E. x  e.  A  y  e.  B  /\  y  e.  C )
)
94, 6, 83bitr4i 280 . . 3  |-  ( E. x  e.  A  y  e.  ( B  i^i  C )  <->  ( y  e. 
U_ x  e.  A  B  /\  y  e.  C
) )
10 eliun 4304 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  i^i  C )  <->  E. x  e.  A  y  e.  ( B  i^i  C ) )
11 elin 3649 . . 3  |-  ( y  e.  ( U_ x  e.  A  B  i^i  C )  <->  ( y  e. 
U_ x  e.  A  B  /\  y  e.  C
) )
129, 10, 113bitr4i 280 . 2  |-  ( y  e.  U_ x  e.  A  ( B  i^i  C )  <->  y  e.  (
U_ x  e.  A  B  i^i  C ) )
1312eqriv 2418 1  |-  U_ x  e.  A  ( B  i^i  C )  =  (
U_ x  e.  A  B  i^i  C )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    e. wcel 1872   F/_wnfc 2566   E.wrex 2772    i^i cin 3435   U_ciun 4299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082  df-in 3443  df-iun 4301
This theorem is referenced by:  esum2dlem  28921
  Copyright terms: Public domain W3C validator