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Theorem uniiunlem 3540
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 2455 . . . . . 6  |-  ( y  =  z  ->  (
y  =  B  <->  z  =  B ) )
21rexbidv 2848 . . . . 5  |-  ( y  =  z  ->  ( E. x  e.  A  y  =  B  <->  E. x  e.  A  z  =  B ) )
32cbvabv 2594 . . . 4  |-  { y  |  E. x  e.  A  y  =  B }  =  { z  |  E. x  e.  A  z  =  B }
43sseq1i 3480 . . 3  |-  ( { y  |  E. x  e.  A  y  =  B }  C_  C  <->  { z  |  E. x  e.  A  z  =  B }  C_  C )
5 r19.23v 2931 . . . . 5  |-  ( A. x  e.  A  (
z  =  B  -> 
z  e.  C )  <-> 
( E. x  e.  A  z  =  B  ->  z  e.  C
) )
65albii 1611 . . . 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 3089 . . . 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 3521 . . . 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 1674 . . . . 5  |-  F/ z  B  e.  C
12 eleq1 2523 . . . . 5  |-  ( z  =  B  ->  (
z  e.  C  <->  B  e.  C ) )
1311, 12ceqsalg 3095 . . . 4  |-  ( B  e.  D  ->  ( A. z ( z  =  B  ->  z  e.  C )  <->  B  e.  C ) )
1413ralimi 2811 . . 3  |-  ( A. x  e.  A  B  e.  D  ->  A. x  e.  A  ( A. z ( z  =  B  ->  z  e.  C )  <->  B  e.  C ) )
15 ralbi 2951 . . 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 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 1368    = wceq 1370    e. wcel 1758   {cab 2436   A.wral 2795   E.wrex 2796    C_ wss 3428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-v 3072  df-in 3435  df-ss 3442
This theorem is referenced by:  mreiincl  14638  iunopn  18629  sigaclci  26711  dihglblem5  35251
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