Theorem iunxsng 4067

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 3995	. . 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 2302	. . . 4 |- ( x = A -> ( y e. B <-> y e. C ) )
4	3	rexsng 3730	. . 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 2230	1 |- ( A e. V -> U_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   E.wrex 2521   {csn 3689   U_ciun 3991

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-sn 3695  df-iun 3993

This theorem is referenced by:  iunxsn  4068  iunxprg  4072  rdgisuc1  6615  oasuc  6697  omsuc  6705