MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniiunlem Structured version   Unicode version

Theorem uniiunlem 3526
Description: A subset relationship useful for converting union to indexed union using dfiun2 4304 or dfiun2g 4302 and intersection to indexed intersection using dfiin2 4305. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
Assertion
Ref Expression
uniiunlem  |-  ( A. x  e.  A  B  e.  D  ->  ( A. x  e.  A  B  e.  C  <->  { y  |  E. x  e.  A  y  =  B }  C_  C
) )
Distinct variable groups:    x, y    y, A    y, B    x, C
Allowed substitution hints:    A( x)    B( x)    C( y)    D( x, y)

Proof of Theorem uniiunlem
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2406 . . . . . 6  |-  ( y  =  z  ->  (
y  =  B  <->  z  =  B ) )
21rexbidv 2917 . . . . 5  |-  ( y  =  z  ->  ( E. x  e.  A  y  =  B  <->  E. x  e.  A  z  =  B ) )
32cbvabv 2545 . . . 4  |-  { y  |  E. x  e.  A  y  =  B }  =  { z  |  E. x  e.  A  z  =  B }
43sseq1i 3465 . . 3  |-  ( { y  |  E. x  e.  A  y  =  B }  C_  C  <->  { z  |  E. x  e.  A  z  =  B }  C_  C )
5 r19.23v 2883 . . . . 5  |-  ( A. x  e.  A  (
z  =  B  -> 
z  e.  C )  <-> 
( E. x  e.  A  z  =  B  ->  z  e.  C
) )
65albii 1661 . . . 4  |-  ( A. z A. x  e.  A  ( z  =  B  ->  z  e.  C
)  <->  A. z ( E. x  e.  A  z  =  B  ->  z  e.  C ) )
7 ralcom4 3077 . . . 4  |-  ( A. x  e.  A  A. z ( z  =  B  ->  z  e.  C )  <->  A. z A. x  e.  A  ( z  =  B  ->  z  e.  C
) )
8 abss 3507 . . . 4  |-  ( { z  |  E. x  e.  A  z  =  B }  C_  C  <->  A. z
( E. x  e.  A  z  =  B  ->  z  e.  C
) )
96, 7, 83bitr4i 277 . . 3  |-  ( A. x  e.  A  A. z ( z  =  B  ->  z  e.  C )  <->  { z  |  E. x  e.  A  z  =  B }  C_  C )
104, 9bitr4i 252 . 2  |-  ( { y  |  E. x  e.  A  y  =  B }  C_  C  <->  A. x  e.  A  A. z
( z  =  B  ->  z  e.  C
) )
11 nfv 1728 . . . . 5  |-  F/ z  B  e.  C
12 eleq1 2474 . . . . 5  |-  ( z  =  B  ->  (
z  e.  C  <->  B  e.  C ) )
1311, 12ceqsalg 3083 . . . 4  |-  ( B  e.  D  ->  ( A. z ( z  =  B  ->  z  e.  C )  <->  B  e.  C ) )
1413ralimi 2796 . . 3  |-  ( A. x  e.  A  B  e.  D  ->  A. x  e.  A  ( A. z ( z  =  B  ->  z  e.  C )  <->  B  e.  C ) )
15 ralbi 2937 . . 3  |-  ( A. x  e.  A  ( A. z ( z  =  B  ->  z  e.  C )  <->  B  e.  C )  ->  ( A. x  e.  A  A. z ( z  =  B  ->  z  e.  C )  <->  A. x  e.  A  B  e.  C ) )
1614, 15syl 17 . 2  |-  ( A. x  e.  A  B  e.  D  ->  ( A. x  e.  A  A. z ( z  =  B  ->  z  e.  C )  <->  A. x  e.  A  B  e.  C ) )
1710, 16syl5rbb 258 1  |-  ( A. x  e.  A  B  e.  D  ->  ( A. x  e.  A  B  e.  C  <->  { y  |  E. x  e.  A  y  =  B }  C_  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1403    = wceq 1405    e. wcel 1842   {cab 2387   A.wral 2753   E.wrex 2754    C_ wss 3413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-v 3060  df-in 3420  df-ss 3427
This theorem is referenced by:  mreiincl  15102  iunopn  19591  sigaclci  28460  dihglblem5  34299
  Copyright terms: Public domain W3C validator