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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 4646 | . 2 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
2 | rexcom4 2577 | . 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 3549 | . . . . . 6 ⊢ (𝑥 = 𝐶 → 〈𝑥, 𝑦〉 = 〈𝐶, 𝑦〉) | |
5 | 4 | eqeq2d 2051 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝐶, 𝑦〉)) |
6 | 4 | eleq1d 2106 | . . . . 5 ⊢ (𝑥 = 𝐶 → (〈𝑥, 𝑦〉 ∈ 𝐵 ↔ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
7 | 5, 6 | anbi12d 442 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ (𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵))) |
8 | 3, 7 | rexsn 3415 | . . 3 ⊢ (∃𝑥 ∈ {𝐶} (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ (𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
9 | 8 | exbii 1496 | . 2 ⊢ (∃𝑦∃𝑥 ∈ {𝐶} (𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ ∃𝑦(𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
10 | 1, 2, 9 | 3bitri 195 | 1 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = 〈𝐶, 𝑦〉 ∧ 〈𝐶, 𝑦〉 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∃wrex 2307 Vcvv 2557 {csn 3375 〈cop 3378 ↾ cres 4347 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-xp 4351 df-rel 4352 df-res 4357 |
This theorem is referenced by: (None) |
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