Theorem iinxsng 4600

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 4523	. 2 |- |^|_ x e. { A } B = { y | A. x e. { A } y e. B }
2		iinxsng.1	. . . . 5 |- ( x = A -> B = C )
3	2	eleq2d 2687	. . . 4 |- ( x = A -> ( y e. B <-> y e. C ) )
4	3	ralsng 4218	. . 3 |- ( A e. V -> ( A. x e. { A } y e. B <-> y e. C ) )
5	4	abbi1dv 2743	. 2 |- ( A e. V -> { y | A. x e. { A } y e. B } = C )
6	1, 5	syl5eq 2668	1 |- ( A e. V -> |^|_ x e. { A } B = C )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   {csn 4177   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-sbc 3436  df-sn 4178  df-iin 4523

This theorem is referenced by:  polatN  35217