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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 4741 | . 2 ⊢ Rel (𝐴 ↾ {𝐵}) | |
2 | relxp 4547 | . 2 ⊢ Rel ({𝐵} × (𝐴 “ {𝐵})) | |
3 | ancom 262 | . . . 4 ⊢ ((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
4 | vex 2622 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
5 | vex 2622 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | elimasn 4799 | . . . . . 6 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) |
7 | elsni 3464 | . . . . . . . . 9 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
8 | 7 | sneqd 3459 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → {𝑥} = {𝐵}) |
9 | 8 | imaeq2d 4774 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵})) |
10 | 9 | eleq2d 2157 | . . . . . 6 ⊢ (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
11 | 6, 10 | syl5bbr 192 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
12 | 11 | pm5.32i 442 | . . . 4 ⊢ ((𝑥 ∈ {𝐵} ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) |
13 | 3, 12 | bitri 182 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) |
14 | 5 | opelres 4718 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ {𝐵}) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ {𝐵})) |
15 | opelxp 4467 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) | |
16 | 13, 14, 15 | 3bitr4i 210 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ {𝐵}) ↔ 〈𝑥, 𝑦〉 ∈ ({𝐵} × (𝐴 “ {𝐵}))) |
17 | 1, 2, 16 | eqrelriiv 4532 | 1 ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1289 ∈ wcel 1438 {csn 3446 〈cop 3449 × cxp 4436 ↾ cres 4440 “ cima 4441 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-sbc 2841 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-opab 3900 df-xp 4444 df-rel 4445 df-cnv 4446 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 |
This theorem is referenced by: (None) |
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