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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 6001 | . 2 ⊢ Rel (𝐴 ↾ {𝐵}) | |
2 | relxp 5685 | . 2 ⊢ Rel ({𝐵} × (𝐴 “ {𝐵})) | |
3 | vex 3470 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3470 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | elimasn 6079 | . . . . 5 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴) |
6 | elsni 4638 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
7 | 6 | sneqd 4633 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐵} → {𝑥} = {𝐵}) |
8 | 7 | imaeq2d 6050 | . . . . . 6 ⊢ (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵})) |
9 | 8 | eleq2d 2811 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
10 | 5, 9 | bitr3id 285 | . . . 4 ⊢ (𝑥 ∈ {𝐵} → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
11 | 10 | pm5.32i 574 | . . 3 ⊢ ((𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) |
12 | 4 | opelresi 5980 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)) |
13 | opelxp 5703 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) | |
14 | 11, 12, 13 | 3bitr4i 303 | . 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐵} × (𝐴 “ {𝐵}))) |
15 | 1, 2, 14 | eqrelriiv 5781 | 1 ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 {csn 4621 ⟨cop 4627 × cxp 5665 ↾ cres 5669 “ cima 5670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-xp 5673 df-rel 5674 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 |
This theorem is referenced by: gsum2dlem2 19887 dprd2da 19960 ustneism 24072 |
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