Theorem iunxsngf 3943

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 3870	. . 3 |- ( y e. U_ x e. { A } B <-> E. x e. { A } y e. B )
2		rexsns 3615	. . . 4 |- ( E. x e. { A } y e. B <-> [. A / x ]. y e. B )
3		iunxsngf.1	. . . . . 6 |- F/_ x C
4	3	nfcri 2302	. . . . 5 |- F/ x y e. C
5		iunxsngf.2	. . . . . 6 |- ( x = A -> B = C )
6	5	eleq2d 2236	. . . . 5 |- ( x = A -> ( y e. B <-> y e. C ) )
7	4, 6	sbciegf 2982	. . . 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 2163	1 |- ( A e. V -> U_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   F/_wnfc 2295   E.wrex 2445   [.wsbc 2951   {csn 3576   U_ciun 3866

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-sn 3582  df-iun 3868

This theorem is referenced by:  iunfidisj  6911  iuncld  12755