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Theorem uniiunlem 3375
Description: A subset relationship useful for converting union to indexed union using dfiun2 4068 or dfiun2g 4066 and intersection to indexed intersection using dfiin2 4069. (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 2394 . . . . . 6  |-  ( y  =  z  ->  (
y  =  B  <->  z  =  B ) )
21rexbidv 2671 . . . . 5  |-  ( y  =  z  ->  ( E. x  e.  A  y  =  B  <->  E. x  e.  A  z  =  B ) )
32cbvabv 2507 . . . 4  |-  { y  |  E. x  e.  A  y  =  B }  =  { z  |  E. x  e.  A  z  =  B }
43sseq1i 3316 . . 3  |-  ( { y  |  E. x  e.  A  y  =  B }  C_  C  <->  { z  |  E. x  e.  A  z  =  B }  C_  C )
5 r19.23v 2766 . . . . 5  |-  ( A. x  e.  A  (
z  =  B  -> 
z  e.  C )  <-> 
( E. x  e.  A  z  =  B  ->  z  e.  C
) )
65albii 1572 . . . 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 2918 . . . 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 3356 . . . 4  |-  ( { z  |  E. x  e.  A  z  =  B }  C_  C  <->  A. z
( E. x  e.  A  z  =  B  ->  z  e.  C
) )
96, 7, 83bitr4i 269 . . 3  |-  ( A. x  e.  A  A. z ( z  =  B  ->  z  e.  C )  <->  { z  |  E. x  e.  A  z  =  B }  C_  C )
104, 9bitr4i 244 . 2  |-  ( { y  |  E. x  e.  A  y  =  B }  C_  C  <->  A. x  e.  A  A. z
( z  =  B  ->  z  e.  C
) )
11 nfv 1626 . . . . 5  |-  F/ z  B  e.  C
12 eleq1 2448 . . . . 5  |-  ( z  =  B  ->  (
z  e.  C  <->  B  e.  C ) )
1311, 12ceqsalg 2924 . . . 4  |-  ( B  e.  D  ->  ( A. z ( z  =  B  ->  z  e.  C )  <->  B  e.  C ) )
1413ralimi 2725 . . 3  |-  ( A. x  e.  A  B  e.  D  ->  A. x  e.  A  ( A. z ( z  =  B  ->  z  e.  C )  <->  B  e.  C ) )
15 ralbi 2786 . . 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 16 . 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 250 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 set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1717   {cab 2374   A.wral 2650   E.wrex 2651    C_ wss 3264
This theorem is referenced by:  mreiincl  13749  iunopn  16895  sigaclci  24312  dihglblem5  31414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-v 2902  df-in 3271  df-ss 3278
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