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Theorem iinxsng 3786
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 3716 . 2 𝑥 ∈ {𝐴}𝐵 = {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦𝐵}
2 iinxsng.1 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
32eleq2d 2154 . . . 4 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
43ralsng 3466 . . 3 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
54abbi1dv 2204 . 2 (𝐴𝑉 → {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦𝐵} = 𝐶)
61, 5syl5eq 2129 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  {cab 2071  wral 2355  {csn 3431   ciin 3714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-sbc 2830  df-sn 3437  df-iin 3716
This theorem is referenced by: (None)
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