Theorem iinxsng 3894

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 3824	. 2 |- |^|_ x e. { A } B = { y | A. x e. { A } y e. B }
2		iinxsng.1	. . . . 5 |- ( x = A -> B = C )
3	2	eleq2d 2210	. . . 4 |- ( x = A -> ( y e. B <-> y e. C ) )
4	3	ralsng 3571	. . 3 |- ( A e. V -> ( A. x e. { A } y e. B <-> y e. C ) )
5	4	abbi1dv 2260	. 2 |- ( A e. V -> { y | A. x e. { A } y e. B } = C )
6	1, 5	syl5eq 2185	1 |- ( A e. V -> |^|_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   {cab 2126   A.wral 2417   {csn 3532   |^|_ciin 3822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-sbc 2914  df-sn 3538  df-iin 3824

This theorem is referenced by: (None)