Theorem iinxsng 3986

Description: A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Hypothesis

Ref	Expression
iinxsng.1	|- ( x = A -> B = C )

Assertion

Ref	Expression
iinxsng	|- ( A e. V -> |^|_ x e. { A } B = C )

Distinct variable groups:   x, A   x, C

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

Proof of Theorem iinxsng

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 3915	. 2 |- |^|_ x e. { A } B = { y | A. x e. { A } y e. B }
2		iinxsng.1	. . . . 5 |- ( x = A -> B = C )
3	2	eleq2d 2263	. . . 4 |- ( x = A -> ( y e. B <-> y e. C ) )
4	3	ralsng 3658	. . 3 |- ( A e. V -> ( A. x e. { A } y e. B <-> y e. C ) )
5	4	abbi1dv 2313	. 2 |- ( A e. V -> { y | A. x e. { A } y e. B } = C )
6	1, 5	eqtrid 2238	1 |- ( A e. V -> |^|_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   {csn 3618   |^|_ciin 3913

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-sbc 2986  df-sn 3624  df-iin 3915

This theorem is referenced by: (None)