Theorem iunxsng 3854

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 3783	. . 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 2184	. . . 4 |- ( x = A -> ( y e. B <-> y e. C ) )
4	3	rexsng 3531	. . 3 |- ( A e. V -> ( E. x e. { A } y e. B <-> y e. C ) )
5	1, 4	syl5bb 191	. 2 |- ( A e. V -> ( y e. U_ x e. { A } B <-> y e. C ) )
6	5	eqrdv 2113	1 |- ( A e. V -> U_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1314   e. wcel 1463   E.wrex 2391   {csn 3493   U_ciun 3779

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-sn 3499  df-iun 3781

This theorem is referenced by:  iunxsn  3855  iunxprg  3859  rdgisuc1  6235  oasuc  6314  omsuc  6322