Theorem iunxsn 4087

Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)

Hypotheses

Ref	Expression
iunxsn.1	|- A e. _V
iunxsn.2	|- ( x = A -> B = C )

Assertion

Ref	Expression
iunxsn	|- U_ x e. { A } B = C

Distinct variable groups:   x, A   x, C

Allowed substitution hint:   B( x)

Proof of Theorem iunxsn

Step	Hyp	Ref	Expression
1		iunxsn.1	. 2 |- A e. _V
2		iunxsn.2	. . 3 |- ( x = A -> B = C )
3	2	iunxsng 4086	. 2 |- ( A e. _V -> U_ x e. { A } B = C )
4	1, 3	ax-mp 5	1 |- U_ x e. { A } B = C

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3708   U_ciun 4010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-sn 3714  df-iun 4012

This theorem is referenced by:  iunsuc  4563  fsum2dlemstep  12182  fsumiun  12225  fprod2dlemstep  12370