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Theorem iunxsngf2 39230
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
iunxsngf2.1  |-  F/_ x C
iunxsngf2.2  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
iunxsngf2  |-  ( A  e.  V  ->  U_ x  e.  { A } B  =  C )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    C( x)    V( x)

Proof of Theorem iunxsngf2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eliun 4524 . . 3  |-  ( y  e.  U_ x  e. 
{ A } B  <->  E. x  e.  { A } y  e.  B
)
2 iunxsngf2.1 . . . . 5  |-  F/_ x C
32nfcri 2758 . . . 4  |-  F/ x  y  e.  C
4 iunxsngf2.2 . . . . 5  |-  ( x  =  A  ->  B  =  C )
54eleq2d 2687 . . . 4  |-  ( x  =  A  ->  (
y  e.  B  <->  y  e.  C ) )
63, 5rexsngf 39220 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  { A } y  e.  B  <->  y  e.  C ) )
71, 6syl5bb 272 . 2  |-  ( A  e.  V  ->  (
y  e.  U_ x  e.  { A } B  <->  y  e.  C ) )
87eqrdv 2620 1  |-  ( A  e.  V  ->  U_ x  e.  { A } B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1483    e. wcel 1990   F/_wnfc 2751   E.wrex 2913   {csn 4177   U_ciun 4520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-sn 4178  df-iun 4522
This theorem is referenced by:  iunxsnf  39233
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