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| Description: Membership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.) | 
| Ref | Expression | 
|---|---|
| elsnres.1 | ⊢ 𝐶 ∈ V | 
| Ref | Expression | 
|---|---|
| elsnres | ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elres 6037 | . 2 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
| 2 | rexcom4 3287 | . 2 ⊢ (∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ ∃𝑦∃𝑥 ∈ {𝐶} (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
| 3 | elsnres.1 | . . . 4 ⊢ 𝐶 ∈ V | |
| 4 | opeq1 4872 | . . . . . 6 ⊢ (𝑥 = 𝐶 → 〈𝑥, 𝑦〉 = 〈𝐶, 𝑦〉) | |
| 5 | 4 | eqeq2d 2747 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝐶, 𝑦〉)) | 
| 6 | 4 | eleq1d 2825 | . . . . 5 ⊢ (𝑥 = 𝐶 → (〈𝑥, 𝑦〉 ∈ 𝐵 ↔ 〈𝐶, 𝑦〉 ∈ 𝐵)) | 
| 7 | 5, 6 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ (𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵))) | 
| 8 | 3, 7 | rexsn 4681 | . . 3 ⊢ (∃𝑥 ∈ {𝐶} (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ (𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) | 
| 9 | 8 | exbii 1847 | . 2 ⊢ (∃𝑦∃𝑥 ∈ {𝐶} (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ ∃𝑦(𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) | 
| 10 | 1, 2, 9 | 3bitri 297 | 1 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ∃wrex 3069 Vcvv 3479 {csn 4625 〈cop 4631 ↾ cres 5686 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-res 5696 | 
| This theorem is referenced by: fvn0ssdmfun 7093 frxp 8152 gsumhashmul 33065 | 
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