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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 4850 | . 2 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
2 | rexcom4 2704 | . 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 3700 | . . . . . 6 ⊢ (𝑥 = 𝐶 → 〈𝑥, 𝑦〉 = 〈𝐶, 𝑦〉) | |
5 | 4 | eqeq2d 2149 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝐶, 𝑦〉)) |
6 | 4 | eleq1d 2206 | . . . . 5 ⊢ (𝑥 = 𝐶 → (〈𝑥, 𝑦〉 ∈ 𝐵 ↔ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
7 | 5, 6 | anbi12d 464 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ (𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵))) |
8 | 3, 7 | rexsn 3563 | . . 3 ⊢ (∃𝑥 ∈ {𝐶} (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ (𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
9 | 8 | exbii 1584 | . 2 ⊢ (∃𝑦∃𝑥 ∈ {𝐶} (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ ∃𝑦(𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
10 | 1, 2, 9 | 3bitri 205 | 1 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∃wrex 2415 Vcvv 2681 {csn 3522 〈cop 3525 ↾ cres 4536 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 df-xp 4540 df-rel 4541 df-res 4546 |
This theorem is referenced by: (None) |
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