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Theorem iunxsngf 3990
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 3916 . . 3 (𝑦 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦𝐵)
2 rexsns 3657 . . . 4 (∃𝑥 ∈ {𝐴}𝑦𝐵[𝐴 / 𝑥]𝑦𝐵)
3 iunxsngf.1 . . . . . 6 𝑥𝐶
43nfcri 2330 . . . . 5 𝑥 𝑦𝐶
5 iunxsngf.2 . . . . . 6 (𝑥 = 𝐴𝐵 = 𝐶)
65eleq2d 2263 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
74, 6sbciegf 3017 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐶))
82, 7bitrid 192 . . 3 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
91, 8bitrid 192 . 2 (𝐴𝑉 → (𝑦 𝑥 ∈ {𝐴}𝐵𝑦𝐶))
109eqrdv 2191 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2164  wnfc 2323  wrex 2473  [wsbc 2985  {csn 3618   ciun 3912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-sn 3624  df-iun 3914
This theorem is referenced by:  iunfidisj  7005  iuncld  14283
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