Theorem iunxsngf 29375

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 4524	. . 3 |- ( y e. U_ x e. { A } B <-> E. x e. { A } y e. B )
2		rexsns 4217	. . . 4 |- ( E. x e. { A } y e. B <-> [. A / x ]. y e. B )
3		iunxsngf.1	. . . . . 6 |- F/_ x C
4	3	nfcri 2758	. . . . 5 |- F/ x y e. C
5		iunxsngf.2	. . . . . 6 |- ( x = A -> B = C )
6	5	eleq2d 2687	. . . . 5 |- ( x = A -> ( y e. B <-> y e. C ) )
7	4, 6	sbciegf 3467	. . . 4 |- ( A e. V -> ( [. A / x ]. y e. B <-> y e. C ) )
8	2, 7	syl5bb 272	. . 3 |- ( A e. V -> ( E. x e. { A } y e. B <-> y e. C ) )
9	1, 8	syl5bb 272	. 2 |- ( A e. V -> ( y e. U_ x e. { A } B <-> y e. C ) )
10	9	eqrdv 2620	1 |- ( A e. V -> U_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   F/_wnfc 2751   E.wrex 2913   [.wsbc 3435   {csn 4177   U_ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-sn 4178  df-iun 4522

This theorem is referenced by:  esum2dlem  30154  fiunelros  30237