Theorem iunxsn 3889

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 3888	. 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 1331   e. wcel 1480   _Vcvv 2686   {csn 3527   U_ciun 3813

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-sn 3533  df-iun 3815

This theorem is referenced by:  iunsuc  4342  fsum2dlemstep  11203  fsumiun  11246