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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 5663 | . 2 ⊢ Rel (𝐴 ↾ {𝐵}) | |
2 | relxp 5361 | . 2 ⊢ Rel ({𝐵} × (𝐴 “ {𝐵})) | |
3 | vex 3418 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3418 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | elimasn 5732 | . . . . 5 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) |
6 | elsni 4415 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
7 | 6 | sneqd 4410 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐵} → {𝑥} = {𝐵}) |
8 | 7 | imaeq2d 5708 | . . . . . 6 ⊢ (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵})) |
9 | 8 | eleq2d 2893 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
10 | 5, 9 | syl5bbr 277 | . . . 4 ⊢ (𝑥 ∈ {𝐵} → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
11 | 10 | pm5.32i 572 | . . 3 ⊢ ((𝑥 ∈ {𝐵} ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) |
12 | 4 | opelresi 5638 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)) |
13 | opelxp 5379 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) | |
14 | 11, 12, 13 | 3bitr4i 295 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ {𝐵}) ↔ 〈𝑥, 𝑦〉 ∈ ({𝐵} × (𝐴 “ {𝐵}))) |
15 | 1, 2, 14 | eqrelriiv 5449 | 1 ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1658 ∈ wcel 2166 {csn 4398 〈cop 4404 × cxp 5341 ↾ cres 5345 “ cima 5346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-xp 5349 df-rel 5350 df-cnv 5351 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 |
This theorem is referenced by: gsum2dlem2 18724 dprd2da 18796 ustneism 22398 |
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