Theorem iunxsng 4083

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

Hypothesis

Ref	Expression
iunxsng.1	|- ( x = A -> B = C )

Assertion

Ref	Expression
iunxsng	|- ( A e. V -> U_ x e. { A } B = C )

Distinct variable groups:   x, A   x, C

Allowed substitution hints:   B( x)   V( x)

Proof of Theorem iunxsng

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4011	. . 3 |- ( y e. U_ x e. { A } B <-> E. x e. { A } y e. B )
2		iunxsng.1	. . . . 5 |- ( x = A -> B = C )
3	2	eleq2d 2308	. . . 4 |- ( x = A -> ( y e. B <-> y e. C ) )
4	3	rexsng 3746	. . 3 |- ( A e. V -> ( E. x e. { A } y e. B <-> y e. C ) )
5	1, 4	bitrid 192	. 2 |- ( A e. V -> ( y e. U_ x e. { A } B <-> y e. C ) )
6	5	eqrdv 2236	1 |- ( A e. V -> U_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   E.wrex 2529   {csn 3705   U_ciun 4007

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 3711  df-iun 4009

This theorem is referenced by:  iunxsn  4084  iunxprg  4088  rdgisuc1  6645  oasuc  6727  omsuc  6735