Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
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 34991 and bj-restsnid 34993. (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 5324 | . . . 4 ⊢ {𝑌} ∈ V | |
2 | elrest 16932 | . . . 4 ⊢ (({𝑌} ∈ V ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴))) | |
3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ 𝑊 → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴))) |
4 | velsn 4557 | . . . . 5 ⊢ (𝑥 ∈ {(𝑦 ∩ 𝐴)} ↔ 𝑥 = (𝑦 ∩ 𝐴)) | |
5 | ineq1 4120 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∩ 𝐴) = (𝑌 ∩ 𝐴)) | |
6 | 5 | sneqd 4553 | . . . . . 6 ⊢ (𝑦 = 𝑌 → {(𝑦 ∩ 𝐴)} = {(𝑌 ∩ 𝐴)}) |
7 | 6 | eleq2d 2823 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑥 ∈ {(𝑦 ∩ 𝐴)} ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)})) |
8 | 4, 7 | bitr3id 288 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)})) |
9 | 8 | rexsng 4590 | . . 3 ⊢ (𝑌 ∈ 𝑉 → (∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)})) |
10 | 3, 9 | sylan9bbr 514 | . 2 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)})) |
11 | 10 | eqrdv 2735 | 1 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃wrex 3062 Vcvv 3408 ∩ cin 3865 {csn 4541 (class class class)co 7213 ↾t crest 16925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-rest 16927 |
This theorem is referenced by: bj-restsnss 34989 bj-restsnss2 34990 |
Copyright terms: Public domain | W3C validator |