Theorem iinxsng 3751

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 3681	. 2 |- |^|_ x e. { A } B = { y | A. x e. { A } y e. B }
2		iinxsng.1	. . . . 5 |- ( x = A -> B = C )
3	2	eleq2d 2148	. . . 4 |- ( x = A -> ( y e. B <-> y e. C ) )
4	3	ralsng 3433	. . 3 |- ( A e. V -> ( A. x e. { A } y e. B <-> y e. C ) )
5	4	abbi1dv 2198	. 2 |- ( A e. V -> { y | A. x e. { A } y e. B } = C )
6	1, 5	syl5eq 2125	1 |- ( A e. V -> |^|_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   {csn 3398   |^|_ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-sbc 2816  df-sn 3404  df-iin 3681

This theorem is referenced by: (None)