Theorem iunxsngf 3898

Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry Arnoux, 2-May-2020.)

Hypotheses

Ref	Expression
iunxsngf.1	|- F/_ x C
iunxsngf.2	|- ( x = A -> B = C )

Assertion

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

Distinct variable group:   x, A

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

Proof of Theorem iunxsngf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 3825	. . 3 |- ( y e. U_ x e. { A } B <-> E. x e. { A } y e. B )
2		rexsns 3570	. . . 4 |- ( E. x e. { A } y e. B <-> [. A / x ]. y e. B )
3		iunxsngf.1	. . . . . 6 |- F/_ x C
4	3	nfcri 2276	. . . . 5 |- F/ x y e. C
5		iunxsngf.2	. . . . . 6 |- ( x = A -> B = C )
6	5	eleq2d 2210	. . . . 5 |- ( x = A -> ( y e. B <-> y e. C ) )
7	4, 6	sbciegf 2944	. . . 4 |- ( A e. V -> ( [. A / x ]. y e. B <-> y e. C ) )
8	2, 7	syl5bb 191	. . 3 |- ( A e. V -> ( E. x e. { A } y e. B <-> y e. C ) )
9	1, 8	syl5bb 191	. 2 |- ( A e. V -> ( y e. U_ x e. { A } B <-> y e. C ) )
10	9	eqrdv 2138	1 |- ( A e. V -> U_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   F/_wnfc 2269   E.wrex 2418   [.wsbc 2913   {csn 3532   U_ciun 3821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-sn 3538  df-iun 3823

This theorem is referenced by:  iunfidisj  6842  iuncld  12323