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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rest10 | Structured version Visualization version GIF version |
Description: An elementwise intersection on a nonempty family by the empty set is the singleton on the empty set. TODO: this generalizes rest0 23193 and could replace it. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-rest10 | ⊢ (𝑋 ∈ 𝑉 → (𝑋 ≠ ∅ → (𝑋 ↾t ∅) = {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5313 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | elrest 17474 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑥 ∈ (𝑋 ↾t ∅) ↔ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦 ∩ ∅))) | |
3 | 1, 2 | mpan2 691 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑥 ∈ (𝑋 ↾t ∅) ↔ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦 ∩ ∅))) |
4 | in0 4401 | . . . . . . . . 9 ⊢ (𝑦 ∩ ∅) = ∅ | |
5 | 4 | eqeq2i 2748 | . . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ ∅) ↔ 𝑥 = ∅) |
6 | 5 | rexbii 3092 | . . . . . . 7 ⊢ (∃𝑦 ∈ 𝑋 𝑥 = (𝑦 ∩ ∅) ↔ ∃𝑦 ∈ 𝑋 𝑥 = ∅) |
7 | df-rex 3069 | . . . . . . . 8 ⊢ (∃𝑦 ∈ 𝑋 𝑥 = ∅ ↔ ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝑥 = ∅)) | |
8 | 19.41v 1947 | . . . . . . . . 9 ⊢ (∃𝑦(𝑦 ∈ 𝑋 ∧ 𝑥 = ∅) ↔ (∃𝑦 𝑦 ∈ 𝑋 ∧ 𝑥 = ∅)) | |
9 | n0 4359 | . . . . . . . . . . 11 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑋) | |
10 | 9 | bicomi 224 | . . . . . . . . . 10 ⊢ (∃𝑦 𝑦 ∈ 𝑋 ↔ 𝑋 ≠ ∅) |
11 | 10 | anbi1i 624 | . . . . . . . . 9 ⊢ ((∃𝑦 𝑦 ∈ 𝑋 ∧ 𝑥 = ∅) ↔ (𝑋 ≠ ∅ ∧ 𝑥 = ∅)) |
12 | 8, 11 | bitri 275 | . . . . . . . 8 ⊢ (∃𝑦(𝑦 ∈ 𝑋 ∧ 𝑥 = ∅) ↔ (𝑋 ≠ ∅ ∧ 𝑥 = ∅)) |
13 | 7, 12 | bitri 275 | . . . . . . 7 ⊢ (∃𝑦 ∈ 𝑋 𝑥 = ∅ ↔ (𝑋 ≠ ∅ ∧ 𝑥 = ∅)) |
14 | 6, 13 | bitri 275 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝑋 𝑥 = (𝑦 ∩ ∅) ↔ (𝑋 ≠ ∅ ∧ 𝑥 = ∅)) |
15 | 14 | baib 535 | . . . . 5 ⊢ (𝑋 ≠ ∅ → (∃𝑦 ∈ 𝑋 𝑥 = (𝑦 ∩ ∅) ↔ 𝑥 = ∅)) |
16 | 3, 15 | sylan9bb 509 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅) → (𝑥 ∈ (𝑋 ↾t ∅) ↔ 𝑥 = ∅)) |
17 | velsn 4647 | . . . 4 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
18 | 16, 17 | bitr4di 289 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅) → (𝑥 ∈ (𝑋 ↾t ∅) ↔ 𝑥 ∈ {∅})) |
19 | 18 | eqrdv 2733 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅) → (𝑋 ↾t ∅) = {∅}) |
20 | 19 | ex 412 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ≠ ∅ → (𝑋 ↾t ∅) = {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 Vcvv 3478 ∩ cin 3962 ∅c0 4339 {csn 4631 (class class class)co 7431 ↾t crest 17467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rest 17469 |
This theorem is referenced by: bj-rest10b 37072 |
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