Theorem iunxsng 4404

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 4330	. . 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 2537	. . . 4 |- ( x = A -> ( y e. B <-> y e. C ) )
4	3	rexsng 4063	. . 3 |- ( A e. V -> ( E. x e. { A } y e. B <-> y e. C ) )
5	1, 4	syl5bb 257	. 2 |- ( A e. V -> ( y e. U_ x e. { A } B <-> y e. C ) )
6	5	eqrdv 2464	1 |- ( A e. V -> U_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1379   e. wcel 1767   E.wrex 2815   {csn 4027   U_ciun 4325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115  df-sbc 3332  df-sn 4028  df-iun 4327

This theorem is referenced by:  iunxsn  4405  iunxprg  31785