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Theorem iunxsngf 3943
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 3870 . . 3 (𝑦 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦𝐵)
2 rexsns 3615 . . . 4 (∃𝑥 ∈ {𝐴}𝑦𝐵[𝐴 / 𝑥]𝑦𝐵)
3 iunxsngf.1 . . . . . 6 𝑥𝐶
43nfcri 2302 . . . . 5 𝑥 𝑦𝐶
5 iunxsngf.2 . . . . . 6 (𝑥 = 𝐴𝐵 = 𝐶)
65eleq2d 2236 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
74, 6sbciegf 2982 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐶))
82, 7syl5bb 191 . . 3 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
91, 8syl5bb 191 . 2 (𝐴𝑉 → (𝑦 𝑥 ∈ {𝐴}𝐵𝑦𝐶))
109eqrdv 2163 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1343  wcel 2136  wnfc 2295  wrex 2445  [wsbc 2951  {csn 3576   ciun 3866
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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-sn 3582  df-iun 3868
This theorem is referenced by:  iunfidisj  6911  iuncld  12755
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