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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 6014 | . 2 ⊢ Rel (𝐴 ↾ {𝐵}) | |
2 | relxp 5696 | . 2 ⊢ Rel ({𝐵} × (𝐴 “ {𝐵})) | |
3 | vex 3475 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3475 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | elimasn 6093 | . . . . 5 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴) |
6 | elsni 4646 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
7 | 6 | sneqd 4641 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐵} → {𝑥} = {𝐵}) |
8 | 7 | imaeq2d 6063 | . . . . . 6 ⊢ (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵})) |
9 | 8 | eleq2d 2815 | . . . . 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 5993 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ↾ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)) |
13 | opelxp 5714 | . . 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 5792 | 1 ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 {csn 4629 ⟨cop 4635 × cxp 5676 ↾ cres 5680 “ cima 5681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 |
This theorem is referenced by: gsum2dlem2 19925 dprd2da 19998 ustneism 24127 |
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