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Theorem iinxsng 4530
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 4452 . 2 𝑥 ∈ {𝐴}𝐵 = {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦𝐵}
2 iinxsng.1 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
32eleq2d 2672 . . . 4 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
43ralsng 4164 . . 3 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
54abbi1dv 2729 . 2 (𝐴𝑉 → {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦𝐵} = 𝐶)
61, 5syl5eq 2655 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  {cab 2595  wral 2895  {csn 4124   ciin 4450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-v 3174  df-sbc 3402  df-sn 4125  df-iin 4452
This theorem is referenced by:  polatN  34038
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