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Theorem iunxsngf 3885
 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 3812 . . 3 (𝑦 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦𝐵)
2 rexsns 3558 . . . 4 (∃𝑥 ∈ {𝐴}𝑦𝐵[𝐴 / 𝑥]𝑦𝐵)
3 iunxsngf.1 . . . . . 6 𝑥𝐶
43nfcri 2273 . . . . 5 𝑥 𝑦𝐶
5 iunxsngf.2 . . . . . 6 (𝑥 = 𝐴𝐵 = 𝐶)
65eleq2d 2207 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
74, 6sbciegf 2935 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐶))
82, 7syl5bb 191 . . 3 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
91, 8syl5bb 191 . 2 (𝐴𝑉 → (𝑦 𝑥 ∈ {𝐴}𝐵𝑦𝐶))
109eqrdv 2135 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  Ⅎwnfc 2266  ∃wrex 2415  [wsbc 2904  {csn 3522  ∪ ciun 3808 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-sn 3528  df-iun 3810 This theorem is referenced by:  iunfidisj  6827  iuncld  12273
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