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Theorem iunxsngf 4980
 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 2380. (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 4888 . . 3 (𝑦 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦𝐵)
2 iunxsngf.1 . . . . 5 𝑥𝐶
32nfcri 2907 . . . 4 𝑥 𝑦𝐶
4 iunxsngf.2 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
54eleq2d 2838 . . . 4 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
63, 5rexsngf 4568 . . 3 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
71, 6syl5bb 286 . 2 (𝐴𝑉 → (𝑦 𝑥 ∈ {𝐴}𝐵𝑦𝐶))
87eqrdv 2757 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112  Ⅎwnfc 2900  ∃wrex 3072  {csn 4523  ∪ ciun 4884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-sbc 3698  df-sn 4524  df-iun 4886 This theorem is referenced by:  esum2dlem  31580  fiunelros  31662  iunxsnf  42072
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