Theorem iunxsng 3732

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 3661	. . 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 2107	. . . 4 |- ( x = A -> ( y e. B <-> y e. C ) )
4	3	rexsng 3412	. . 3 |- ( A e. V -> ( E. x e. { A } y e. B <-> y e. C ) )
5	1, 4	syl5bb 181	. 2 |- ( A e. V -> ( y e. U_ x e. { A } B <-> y e. C ) )
6	5	eqrdv 2038	1 |- ( A e. V -> U_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   E.wrex 2307   {csn 3375   U_ciun 3657

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-sn 3381  df-iun 3659

This theorem is referenced by:  iunxsn  3733  rdgisuc1  5971  oasuc  6044  omsuc  6051