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Mirrors > Home > ILE Home > Th. List > ressn | GIF version |
Description: Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
ressn | ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 4639 | . 2 ⊢ Rel (𝐴 ↾ {𝐵}) | |
2 | relxp 4447 | . 2 ⊢ Rel ({𝐵} × (𝐴 “ {𝐵})) | |
3 | ancom 253 | . . . 4 ⊢ ((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
4 | vex 2560 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
5 | vex 2560 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | elimasn 4692 | . . . . . 6 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) |
7 | elsni 3393 | . . . . . . . . 9 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
8 | 7 | sneqd 3388 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → {𝑥} = {𝐵}) |
9 | 8 | imaeq2d 4668 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵})) |
10 | 9 | eleq2d 2107 | . . . . . 6 ⊢ (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
11 | 6, 10 | syl5bbr 183 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
12 | 11 | pm5.32i 427 | . . . 4 ⊢ ((𝑥 ∈ {𝐵} ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) |
13 | 3, 12 | bitri 173 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) |
14 | 5 | opelres 4617 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ {𝐵}) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ {𝐵})) |
15 | opelxp 4374 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) | |
16 | 13, 14, 15 | 3bitr4i 201 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ {𝐵}) ↔ 〈𝑥, 𝑦〉 ∈ ({𝐵} × (𝐴 “ {𝐵}))) |
17 | 1, 2, 16 | eqrelriiv 4434 | 1 ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∈ wcel 1393 {csn 3375 〈cop 3378 × cxp 4343 ↾ cres 4347 “ cima 4348 |
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-eu 1903 df-mo 1904 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-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 |
This theorem is referenced by: (None) |
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