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Theorem iunxsngf 28758
 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.)
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 4460 . . 3 (𝑦 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦𝐵)
2 velsn 4141 . . . . . . 7 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
32anbi1i 727 . . . . . 6 ((𝑥 ∈ {𝐴} ∧ 𝑦𝐵) ↔ (𝑥 = 𝐴𝑦𝐵))
43exbii 1764 . . . . 5 (∃𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵) ↔ ∃𝑥(𝑥 = 𝐴𝑦𝐵))
5 df-rex 2902 . . . . 5 (∃𝑥 ∈ {𝐴}𝑦𝐵 ↔ ∃𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵))
6 sbc5 3427 . . . . 5 ([𝐴 / 𝑥]𝑦𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑦𝐵))
74, 5, 63bitr4i 291 . . . 4 (∃𝑥 ∈ {𝐴}𝑦𝐵[𝐴 / 𝑥]𝑦𝐵)
8 nfcv 2751 . . . . . 6 𝑥𝑦
9 iunxsngf.1 . . . . . 6 𝑥𝐶
108, 9nfel 2763 . . . . 5 𝑥 𝑦𝐶
11 iunxsngf.2 . . . . . 6 (𝑥 = 𝐴𝐵 = 𝐶)
1211eleq2d 2673 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
1310, 12sbciegf 3434 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐶))
147, 13syl5bb 271 . . 3 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
151, 14syl5bb 271 . 2 (𝐴𝑉 → (𝑦 𝑥 ∈ {𝐴}𝐵𝑦𝐶))
1615eqrdv 2608 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Ⅎwnfc 2738  ∃wrex 2897  [wsbc 3402  {csn 4125  ∪ ciun 4455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-sn 4126  df-iun 4457 This theorem is referenced by:  esum2dlem  29481  fiunelros  29564
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