Theorem iinxsng 4081

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 4010	. 2 |- |^|_ x e. { A } B = { y | A. x e. { A } y e. B }
2		iinxsng.1	. . . . 5 |- ( x = A -> B = C )
3	2	eleq2d 2308	. . . 4 |- ( x = A -> ( y e. B <-> y e. C ) )
4	3	ralsng 3745	. . 3 |- ( A e. V -> ( A. x e. { A } y e. B <-> y e. C ) )
5	4	abbi1dv 2360	. 2 |- ( A e. V -> { y | A. x e. { A } y e. B } = C )
6	1, 5	eqtrid 2283	1 |- ( A e. V -> |^|_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   {csn 3705   |^|_ciin 4008

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-sbc 3052  df-sn 3711  df-iin 4010

This theorem is referenced by: (None)