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Mirrors > Home > MPE Home > Th. List > ressn | Structured version Visualization version GIF version |
Description: Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
Ref | Expression |
---|---|
ressn | ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 5847 | . 2 ⊢ Rel (𝐴 ↾ {𝐵}) | |
2 | relxp 5537 | . 2 ⊢ Rel ({𝐵} × (𝐴 “ {𝐵})) | |
3 | vex 3444 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3444 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | elimasn 5921 | . . . . 5 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) |
6 | elsni 4542 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
7 | 6 | sneqd 4537 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐵} → {𝑥} = {𝐵}) |
8 | 7 | imaeq2d 5896 | . . . . . 6 ⊢ (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵})) |
9 | 8 | eleq2d 2875 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
10 | 5, 9 | bitr3id 288 | . . . 4 ⊢ (𝑥 ∈ {𝐵} → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
11 | 10 | pm5.32i 578 | . . 3 ⊢ ((𝑥 ∈ {𝐵} ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) |
12 | 4 | opelresi 5826 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)) |
13 | opelxp 5555 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) | |
14 | 11, 12, 13 | 3bitr4i 306 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ {𝐵}) ↔ 〈𝑥, 𝑦〉 ∈ ({𝐵} × (𝐴 “ {𝐵}))) |
15 | 1, 2, 14 | eqrelriiv 5627 | 1 ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 〈cop 4531 × cxp 5517 ↾ cres 5521 “ cima 5522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: gsum2dlem2 19084 dprd2da 19157 ustneism 22829 |
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