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Theorem elsnres 4962
Description: Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)
Hypothesis
Ref Expression
elsnres.1 𝐶 ∈ V
Assertion
Ref Expression
elsnres (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶

Proof of Theorem elsnres
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elres 4961 . 2 (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2 rexcom4 2775 . 2 (∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦𝑥 ∈ {𝐶} (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3 elsnres.1 . . . 4 𝐶 ∈ V
4 opeq1 3793 . . . . . 6 (𝑥 = 𝐶 → ⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝑦⟩)
54eqeq2d 2201 . . . . 5 (𝑥 = 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝐶, 𝑦⟩))
64eleq1d 2258 . . . . 5 (𝑥 = 𝐶 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
75, 6anbi12d 473 . . . 4 (𝑥 = 𝐶 → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵)))
83, 7rexsn 3651 . . 3 (∃𝑥 ∈ {𝐶} (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
98exbii 1616 . 2 (∃𝑦𝑥 ∈ {𝐶} (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
101, 2, 93bitri 206 1 (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  wrex 2469  Vcvv 2752  {csn 3607  cop 3610  cres 4646
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-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-opab 4080  df-xp 4650  df-rel 4651  df-res 4656
This theorem is referenced by: (None)
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