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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 4847 | . 2 ⊢ Rel (𝐴 ↾ {𝐵}) | |
2 | relxp 4648 | . 2 ⊢ Rel ({𝐵} × (𝐴 “ {𝐵})) | |
3 | ancom 264 | . . . 4 ⊢ ((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
4 | vex 2689 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
5 | vex 2689 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | elimasn 4906 | . . . . . 6 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) |
7 | elsni 3545 | . . . . . . . . 9 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
8 | 7 | sneqd 3540 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → {𝑥} = {𝐵}) |
9 | 8 | imaeq2d 4881 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵})) |
10 | 9 | eleq2d 2209 | . . . . . 6 ⊢ (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
11 | 6, 10 | syl5bbr 193 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
12 | 11 | pm5.32i 449 | . . . 4 ⊢ ((𝑥 ∈ {𝐵} ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) |
13 | 3, 12 | bitri 183 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) |
14 | 5 | opelres 4824 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ {𝐵}) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ {𝐵})) |
15 | opelxp 4569 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) | |
16 | 13, 14, 15 | 3bitr4i 211 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ {𝐵}) ↔ 〈𝑥, 𝑦〉 ∈ ({𝐵} × (𝐴 “ {𝐵}))) |
17 | 1, 2, 16 | eqrelriiv 4633 | 1 ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∈ wcel 1480 {csn 3527 〈cop 3530 × cxp 4537 ↾ cres 4541 “ cima 4542 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 |
This theorem is referenced by: (None) |
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