Theorem iinxsng 4043

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 3973	. 2 |- |^|_x e. {A}B = {y | A.x e. {A}y e. B}
2		iinxsng.1	. . . . 5 |- (x = A -> B = C)
3	2	eleq2d 2420	. . . 4 |- (x = A -> (y e. B <-> y e. C))
4	3	ralsng 3766	. . 3 |- (A e. V -> (A.x e. {A}y e. B <-> y e. C))
5	4	abbi1dv 2470	. 2 |- (A e. V -> {y | A.x e. {A}y e. B} = C)
6	1, 5	syl5eq 2397	1 |- (A e. V -> |^|_x e. {A}B = C)

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615  {csn 3738  |^|_ciin 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-sbc 3048  df-sn 3742  df-iin 3973

This theorem is referenced by: (None)