![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 23093 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 5311 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | elrest 17416 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑥 ∈ (𝑋 ↾t ∅) ↔ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦 ∩ ∅))) | |
3 | 1, 2 | mpan2 689 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑥 ∈ (𝑋 ↾t ∅) ↔ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦 ∩ ∅))) |
4 | in0 4395 | . . . . . . . . 9 ⊢ (𝑦 ∩ ∅) = ∅ | |
5 | 4 | eqeq2i 2741 | . . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ ∅) ↔ 𝑥 = ∅) |
6 | 5 | rexbii 3091 | . . . . . . 7 ⊢ (∃𝑦 ∈ 𝑋 𝑥 = (𝑦 ∩ ∅) ↔ ∃𝑦 ∈ 𝑋 𝑥 = ∅) |
7 | df-rex 3068 | . . . . . . . 8 ⊢ (∃𝑦 ∈ 𝑋 𝑥 = ∅ ↔ ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝑥 = ∅)) | |
8 | 19.41v 1945 | . . . . . . . . 9 ⊢ (∃𝑦(𝑦 ∈ 𝑋 ∧ 𝑥 = ∅) ↔ (∃𝑦 𝑦 ∈ 𝑋 ∧ 𝑥 = ∅)) | |
9 | n0 4350 | . . . . . . . . . . 11 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑋) | |
10 | 9 | bicomi 223 | . . . . . . . . . 10 ⊢ (∃𝑦 𝑦 ∈ 𝑋 ↔ 𝑋 ≠ ∅) |
11 | 10 | anbi1i 622 | . . . . . . . . 9 ⊢ ((∃𝑦 𝑦 ∈ 𝑋 ∧ 𝑥 = ∅) ↔ (𝑋 ≠ ∅ ∧ 𝑥 = ∅)) |
12 | 8, 11 | bitri 274 | . . . . . . . 8 ⊢ (∃𝑦(𝑦 ∈ 𝑋 ∧ 𝑥 = ∅) ↔ (𝑋 ≠ ∅ ∧ 𝑥 = ∅)) |
13 | 7, 12 | bitri 274 | . . . . . . 7 ⊢ (∃𝑦 ∈ 𝑋 𝑥 = ∅ ↔ (𝑋 ≠ ∅ ∧ 𝑥 = ∅)) |
14 | 6, 13 | bitri 274 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝑋 𝑥 = (𝑦 ∩ ∅) ↔ (𝑋 ≠ ∅ ∧ 𝑥 = ∅)) |
15 | 14 | baib 534 | . . . . 5 ⊢ (𝑋 ≠ ∅ → (∃𝑦 ∈ 𝑋 𝑥 = (𝑦 ∩ ∅) ↔ 𝑥 = ∅)) |
16 | 3, 15 | sylan9bb 508 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅) → (𝑥 ∈ (𝑋 ↾t ∅) ↔ 𝑥 = ∅)) |
17 | velsn 4648 | . . . 4 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
18 | 16, 17 | bitr4di 288 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅) → (𝑥 ∈ (𝑋 ↾t ∅) ↔ 𝑥 ∈ {∅})) |
19 | 18 | eqrdv 2726 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 ≠ ∅) → (𝑋 ↾t ∅) = {∅}) |
20 | 19 | ex 411 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ≠ ∅ → (𝑋 ↾t ∅) = {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2937 ∃wrex 3067 Vcvv 3473 ∩ cin 3948 ∅c0 4326 {csn 4632 (class class class)co 7426 ↾t crest 17409 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-rest 17411 |
This theorem is referenced by: bj-rest10b 36601 |
Copyright terms: Public domain | W3C validator |