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Theorem iinxsng 5093
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 4999 . 2 𝑥 ∈ {𝐴}𝐵 = {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦𝐵}
2 iinxsng.1 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
32eleq2d 2825 . . . 4 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
43ralsng 4680 . . 3 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
54eqabcdv 2874 . 2 (𝐴𝑉 → {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦𝐵} = 𝐶)
61, 5eqtrid 2787 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {cab 2712  wral 3059  {csn 4631   ciin 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-sn 4632  df-iin 4999
This theorem is referenced by:  polatN  39914
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