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Theorem iunxsng 3757
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
Hypothesis
Ref Expression
iunxsng.1 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
iunxsng (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem iunxsng
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 3686 . . 3 (𝑦 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦𝐵)
2 iunxsng.1 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
32eleq2d 2121 . . . 4 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
43rexsng 3437 . . 3 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
51, 4syl5bb 185 . 2 (𝐴𝑉 → (𝑦 𝑥 ∈ {𝐴}𝐵𝑦𝐶))
65eqrdv 2052 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1257  wcel 1407  wrex 2322  {csn 3400   ciun 3682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-sbc 2785  df-sn 3406  df-iun 3684
This theorem is referenced by:  iunxsn  3758  rdgisuc1  5999  oasuc  6072  omsuc  6079
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