Theorem iunxsn 3990

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 3989	. 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 1364   e. wcel 2164   _Vcvv 2760   {csn 3619   U_ciun 3913

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-sn 3625  df-iun 3915

This theorem is referenced by:  iunsuc  4452  fsum2dlemstep  11580  fsumiun  11623  fprod2dlemstep  11768