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Theorem iinxsng 4974
 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 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
iinxsng (𝐴𝑉 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.
StepHypRef Expression
1 df-iin 4885 . 2 𝑥 ∈ {𝐴}𝐵 = {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦𝐵}
2 iinxsng.1 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
32eleq2d 2875 . . . 4 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
43ralsng 4573 . . 3 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
54abbi1dv 2928 . 2 (𝐴𝑉 → {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦𝐵} = 𝐶)
61, 5syl5eq 2845 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  {csn 4525  ∩ ciin 4883 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-sbc 3721  df-sn 4526  df-iin 4885 This theorem is referenced by:  polatN  37246
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