Theorem iunxsngf 3995

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 3921	. . 3 |- ( y e. U_ x e. { A } B <-> E. x e. { A } y e. B )
2		rexsns 3662	. . . 4 |- ( E. x e. { A } y e. B <-> [. A / x ]. y e. B )
3		iunxsngf.1	. . . . . 6 |- F/_ x C
4	3	nfcri 2333	. . . . 5 |- F/ x y e. C
5		iunxsngf.2	. . . . . 6 |- ( x = A -> B = C )
6	5	eleq2d 2266	. . . . 5 |- ( x = A -> ( y e. B <-> y e. C ) )
7	4, 6	sbciegf 3021	. . . 4 |- ( A e. V -> ( [. A / x ]. y e. B <-> y e. C ) )
8	2, 7	bitrid 192	. . 3 |- ( A e. V -> ( E. x e. { A } y e. B <-> y e. C ) )
9	1, 8	bitrid 192	. 2 |- ( A e. V -> ( y e. U_ x e. { A } B <-> y e. C ) )
10	9	eqrdv 2194	1 |- ( A e. V -> U_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   F/_wnfc 2326   E.wrex 2476   [.wsbc 2989   {csn 3623   U_ciun 3917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-sn 3629  df-iun 3919

This theorem is referenced by:  iunfidisj  7021  iuncld  14435