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 ∈ V → ∩x ∈ {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 ∈ {A}B = {y ∣ ∀x ∈ {A}y ∈ B}
2		iinxsng.1	. . . . 5 ⊢ (x = A → B = C)
3	2	eleq2d 2420	. . . 4 ⊢ (x = A → (y ∈ B ↔ y ∈ C))
4	3	ralsng 3766	. . 3 ⊢ (A ∈ V → (∀x ∈ {A}y ∈ B ↔ y ∈ C))
5	4	abbi1dv 2470	. 2 ⊢ (A ∈ V → {y ∣ ∀x ∈ {A}y ∈ B} = C)
6	1, 5	syl5eq 2397	1 ⊢ (A ∈ V → ∩x ∈ {A}B = C)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀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)