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Theorem iunxsngf 5085
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry Arnoux, 2-May-2020.) Avoid ax-13 2363. (Revised by Gino Giotto, 19-May-2023.)
Hypotheses
Ref Expression
iunxsngf.1 𝑥𝐶
iunxsngf.2 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
iunxsngf (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem iunxsngf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 4991 . . 3 (𝑦 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦𝐵)
2 iunxsngf.1 . . . . 5 𝑥𝐶
32nfcri 2882 . . . 4 𝑥 𝑦𝐶
4 iunxsngf.2 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
54eleq2d 2811 . . . 4 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
63, 5rexsngf 4666 . . 3 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
71, 6bitrid 283 . 2 (𝐴𝑉 → (𝑦 𝑥 ∈ {𝐴}𝐵𝑦𝐶))
87eqrdv 2722 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wnfc 2875  wrex 3062  {csn 4620   ciun 4987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-rex 3063  df-v 3468  df-sbc 3770  df-sn 4621  df-iun 4989
This theorem is referenced by:  esum2dlem  33545  fiunelros  33627  iunxsnf  44205
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