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Theorem uniiunlem 3260
Description: A subset relationship useful for converting union to indexed union using dfiun2 3937 or dfiun2g 3935 and intersection to indexed intersection using dfiin2 3938. (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 2289 . . . . . 6  |-  ( y  =  z  ->  (
y  =  B  <->  z  =  B ) )
21rexbidv 2564 . . . . 5  |-  ( y  =  z  ->  ( E. x  e.  A  y  =  B  <->  E. x  e.  A  z  =  B ) )
32cbvabv 2402 . . . 4  |-  { y  |  E. x  e.  A  y  =  B }  =  { z  |  E. x  e.  A  z  =  B }
43sseq1i 3202 . . 3  |-  ( { y  |  E. x  e.  A  y  =  B }  C_  C  <->  { z  |  E. x  e.  A  z  =  B }  C_  C )
5 r19.23v 2659 . . . . 5  |-  ( A. x  e.  A  (
z  =  B  -> 
z  e.  C )  <-> 
( E. x  e.  A  z  =  B  ->  z  e.  C
) )
65albii 1553 . . . 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 2806 . . . 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 3242 . . . 4  |-  ( { z  |  E. x  e.  A  z  =  B }  C_  C  <->  A. z
( E. x  e.  A  z  =  B  ->  z  e.  C
) )
96, 7, 83bitr4i 268 . . 3  |-  ( A. x  e.  A  A. z ( z  =  B  ->  z  e.  C )  <->  { z  |  E. x  e.  A  z  =  B }  C_  C )
104, 9bitr4i 243 . 2  |-  ( { y  |  E. x  e.  A  y  =  B }  C_  C  <->  A. x  e.  A  A. z
( z  =  B  ->  z  e.  C
) )
11 nfv 1605 . . . . 5  |-  F/ z  B  e.  C
12 eleq1 2343 . . . . 5  |-  ( z  =  B  ->  (
z  e.  C  <->  B  e.  C ) )
1311, 12ceqsalg 2812 . . . 4  |-  ( B  e.  D  ->  ( A. z ( z  =  B  ->  z  e.  C )  <->  B  e.  C ) )
1413ralimi 2618 . . 3  |-  ( A. x  e.  A  B  e.  D  ->  A. x  e.  A  ( A. z ( z  =  B  ->  z  e.  C )  <->  B  e.  C ) )
15 ralbi 2679 . . 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 15 . 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 249 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 176   A.wal 1527    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    C_ wss 3152
This theorem is referenced by:  mreiincl  13498  iunopn  16644  sigaclci  23493  smbkle  26043  dihglblem5  31488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166
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