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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 34379 and bj-restsnid 34381. (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 5332 | . . . 4 ⊢ {𝑌} ∈ V | |
2 | elrest 16701 | . . . 4 ⊢ (({𝑌} ∈ V ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴))) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐴 ∈ 𝑊 → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴))) |
4 | velsn 4583 | . . . . 5 ⊢ (𝑥 ∈ {(𝑦 ∩ 𝐴)} ↔ 𝑥 = (𝑦 ∩ 𝐴)) | |
5 | ineq1 4181 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∩ 𝐴) = (𝑌 ∩ 𝐴)) | |
6 | 5 | sneqd 4579 | . . . . . 6 ⊢ (𝑦 = 𝑌 → {(𝑦 ∩ 𝐴)} = {(𝑌 ∩ 𝐴)}) |
7 | 6 | eleq2d 2898 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑥 ∈ {(𝑦 ∩ 𝐴)} ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)})) |
8 | 4, 7 | syl5bbr 287 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)})) |
9 | 8 | rexsng 4614 | . . 3 ⊢ (𝑌 ∈ 𝑉 → (∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)})) |
10 | 3, 9 | sylan9bbr 513 | . 2 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)})) |
11 | 10 | eqrdv 2819 | 1 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 ∩ cin 3935 {csn 4567 (class class class)co 7156 ↾t crest 16694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-rest 16696 |
This theorem is referenced by: bj-restsnss 34377 bj-restsnss2 34378 |
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