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Theorem iunxsngf2 39727
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
iunxsngf2.1 𝑥𝐶
iunxsngf2.2 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
iunxsngf2 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem iunxsngf2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 4674 . . 3 (𝑦 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦𝐵)
2 iunxsngf2.1 . . . . 5 𝑥𝐶
32nfcri 2894 . . . 4 𝑥 𝑦𝐶
4 iunxsngf2.2 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
54eleq2d 2823 . . . 4 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
63, 5rexsngf 39717 . . 3 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
71, 6syl5bb 272 . 2 (𝐴𝑉 → (𝑦 𝑥 ∈ {𝐴}𝐵𝑦𝐶))
87eqrdv 2756 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2137  wnfc 2887  wrex 3049  {csn 4319   ciun 4670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-v 3340  df-sbc 3575  df-sn 4320  df-iun 4672
This theorem is referenced by:  iunxsnf  39730
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