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Theorem iunxsngf 3948
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry Arnoux, 2-May-2020.)
Hypotheses
Ref Expression
iunxsngf.1  |-  F/_ x C
iunxsngf.2  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
iunxsngf  |-  ( 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 iunxsngf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eliun 3875 . . 3  |-  ( y  e.  U_ x  e. 
{ A } B  <->  E. x  e.  { A } y  e.  B
)
2 rexsns 3620 . . . 4  |-  ( E. x  e.  { A } y  e.  B  <->  [. A  /  x ]. y  e.  B )
3 iunxsngf.1 . . . . . 6  |-  F/_ x C
43nfcri 2306 . . . . 5  |-  F/ x  y  e.  C
5 iunxsngf.2 . . . . . 6  |-  ( x  =  A  ->  B  =  C )
65eleq2d 2240 . . . . 5  |-  ( x  =  A  ->  (
y  e.  B  <->  y  e.  C ) )
74, 6sbciegf 2986 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  B  <->  y  e.  C ) )
82, 7syl5bb 191 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  { A } y  e.  B  <->  y  e.  C ) )
91, 8syl5bb 191 . 2  |-  ( A  e.  V  ->  (
y  e.  U_ x  e.  { A } B  <->  y  e.  C ) )
109eqrdv 2168 1  |-  ( A  e.  V  ->  U_ x  e.  { A } B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141   F/_wnfc 2299   E.wrex 2449   [.wsbc 2955   {csn 3581   U_ciun 3871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-sn 3587  df-iun 3873
This theorem is referenced by:  iunfidisj  6921  iuncld  12874
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