Theorem iunxsngf 4005

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 3931	. . 3 |- ( y e. U_ x e. { A } B <-> E. x e. { A } y e. B )
2		rexsns 3672	. . . 4 |- ( E. x e. { A } y e. B <-> [. A / x ]. y e. B )
3		iunxsngf.1	. . . . . 6 |- F/_ x C
4	3	nfcri 2342	. . . . 5 |- F/ x y e. C
5		iunxsngf.2	. . . . . 6 |- ( x = A -> B = C )
6	5	eleq2d 2275	. . . . 5 |- ( x = A -> ( y e. B <-> y e. C ) )
7	4, 6	sbciegf 3030	. . . 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 2203	1 |- ( A e. V -> U_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   F/_wnfc 2335   E.wrex 2485   [.wsbc 2998   {csn 3633   U_ciun 3927

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-sn 3639  df-iun 3929

This theorem is referenced by:  iunfidisj  7050  iuncld  14620