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Theorem iunssf 37500
Description: Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypothesis
Ref Expression
iunssf.1  |-  F/_ x C
Assertion
Ref Expression
iunssf  |-  ( U_ x  e.  A  B  C_  C  <->  A. x  e.  A  B  C_  C )

Proof of Theorem iunssf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-iun 4271 . . 3  |-  U_ x  e.  A  B  =  { y  |  E. x  e.  A  y  e.  B }
21sseq1i 3442 . 2  |-  ( U_ x  e.  A  B  C_  C  <->  { y  |  E. x  e.  A  y  e.  B }  C_  C
)
3 abss 3484 . 2  |-  ( { y  |  E. x  e.  A  y  e.  B }  C_  C  <->  A. y
( E. x  e.  A  y  e.  B  ->  y  e.  C ) )
4 dfss2 3407 . . . 4  |-  ( B 
C_  C  <->  A. y
( y  e.  B  ->  y  e.  C ) )
54ralbii 2823 . . 3  |-  ( A. x  e.  A  B  C_  C  <->  A. x  e.  A  A. y ( y  e.  B  ->  y  e.  C ) )
6 ralcom4 3052 . . 3  |-  ( A. x  e.  A  A. y ( y  e.  B  ->  y  e.  C )  <->  A. y A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
7 nfcv 2612 . . . . . 6  |-  F/_ x
y
8 iunssf.1 . . . . . 6  |-  F/_ x C
97, 8nfel 2624 . . . . 5  |-  F/ x  y  e.  C
109r19.23 2862 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  -> 
y  e.  C )  <-> 
( E. x  e.  A  y  e.  B  ->  y  e.  C ) )
1110albii 1699 . . 3  |-  ( A. y A. x  e.  A  ( y  e.  B  ->  y  e.  C )  <->  A. y ( E. x  e.  A  y  e.  B  ->  y  e.  C
) )
125, 6, 113bitrri 280 . 2  |-  ( A. y ( E. x  e.  A  y  e.  B  ->  y  e.  C
)  <->  A. x  e.  A  B  C_  C )
132, 3, 123bitri 279 1  |-  ( U_ x  e.  A  B  C_  C  <->  A. x  e.  A  B  C_  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450    e. wcel 1904   {cab 2457   F/_wnfc 2599   A.wral 2756   E.wrex 2757    C_ wss 3390   U_ciun 4269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-v 3033  df-in 3397  df-ss 3404  df-iun 4271
This theorem is referenced by:  iunmapss  37568
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