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Mirrors > Home > ILE Home > Th. List > elsnres | GIF version |
Description: Memebership 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 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) ↔ ∃𝑦∃𝑥 ∈ {𝐶} (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
3 | elsnres.1 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | opeq1 3793 | . . . . . 6 ⊢ (𝑥 = 𝐶 → 〈𝑥, 𝑦〉 = 〈𝐶, 𝑦〉) | |
5 | 4 | eqeq2d 2201 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝐶, 𝑦〉)) |
6 | 4 | eleq1d 2258 | . . . . 5 ⊢ (𝑥 = 𝐶 → (〈𝑥, 𝑦〉 ∈ 𝐵 ↔ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
7 | 5, 6 | anbi12d 473 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ (𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵))) |
8 | 3, 7 | rexsn 3651 | . . 3 ⊢ (∃𝑥 ∈ {𝐶} (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ (𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
9 | 8 | exbii 1616 | . 2 ⊢ (∃𝑦∃𝑥 ∈ {𝐶} (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ ∃𝑦(𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
10 | 1, 2, 9 | 3bitri 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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