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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-restsn | Structured version Visualization version GIF version |
Description: An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 36473 and bj-restsnid 36475. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-restsn | ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5424 | . . . 4 ⊢ {𝑌} ∈ V | |
2 | elrest 17382 | . . . 4 ⊢ (({𝑌} ∈ V ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴))) | |
3 | 1, 2 | mpan 687 | . . 3 ⊢ (𝐴 ∈ 𝑊 → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴))) |
4 | velsn 4639 | . . . . 5 ⊢ (𝑥 ∈ {(𝑦 ∩ 𝐴)} ↔ 𝑥 = (𝑦 ∩ 𝐴)) | |
5 | ineq1 4200 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∩ 𝐴) = (𝑌 ∩ 𝐴)) | |
6 | 5 | sneqd 4635 | . . . . . 6 ⊢ (𝑦 = 𝑌 → {(𝑦 ∩ 𝐴)} = {(𝑌 ∩ 𝐴)}) |
7 | 6 | eleq2d 2813 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑥 ∈ {(𝑦 ∩ 𝐴)} ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)})) |
8 | 4, 7 | bitr3id 285 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)})) |
9 | 8 | rexsng 4673 | . . 3 ⊢ (𝑌 ∈ 𝑉 → (∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)})) |
10 | 3, 9 | sylan9bbr 510 | . 2 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)})) |
11 | 10 | eqrdv 2724 | 1 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 Vcvv 3468 ∩ cin 3942 {csn 4623 (class class class)co 7405 ↾t crest 17375 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
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-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-rest 17377 |
This theorem is referenced by: bj-restsnss 36471 bj-restsnss2 36472 |
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