Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elhoi | Structured version Visualization version GIF version |
Description: Membership in a multidimensional half-open interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
elhoi.1 | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
elhoi | ⊢ (𝜑 → (𝑌 ∈ ((𝐴[,)𝐵) ↑m 𝑋) ↔ (𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovexd 7180 | . . 3 ⊢ (𝜑 → (𝐴[,)𝐵) ∈ V) | |
2 | elhoi.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | elmapg 8408 | . . 3 ⊢ (((𝐴[,)𝐵) ∈ V ∧ 𝑋 ∈ 𝑉) → (𝑌 ∈ ((𝐴[,)𝐵) ↑m 𝑋) ↔ 𝑌:𝑋⟶(𝐴[,)𝐵))) | |
4 | 1, 2, 3 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑌 ∈ ((𝐴[,)𝐵) ↑m 𝑋) ↔ 𝑌:𝑋⟶(𝐴[,)𝐵))) |
5 | id 22 | . . . . . 6 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → 𝑌:𝑋⟶(𝐴[,)𝐵)) | |
6 | icossxr 12809 | . . . . . . 7 ⊢ (𝐴[,)𝐵) ⊆ ℝ* | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → (𝐴[,)𝐵) ⊆ ℝ*) |
8 | 5, 7 | fssd 6521 | . . . . 5 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → 𝑌:𝑋⟶ℝ*) |
9 | ffvelrn 6841 | . . . . . 6 ⊢ ((𝑌:𝑋⟶(𝐴[,)𝐵) ∧ 𝑥 ∈ 𝑋) → (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) | |
10 | 9 | ralrimiva 3179 | . . . . 5 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) |
11 | 8, 10 | jca 512 | . . . 4 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → (𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))) |
12 | ffn 6507 | . . . . . . 7 ⊢ (𝑌:𝑋⟶ℝ* → 𝑌 Fn 𝑋) | |
13 | 12 | adantr 481 | . . . . . 6 ⊢ ((𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → 𝑌 Fn 𝑋) |
14 | simpr 485 | . . . . . 6 ⊢ ((𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) | |
15 | 13, 14 | jca 512 | . . . . 5 ⊢ ((𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → (𝑌 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))) |
16 | ffnfv 6874 | . . . . 5 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) ↔ (𝑌 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))) | |
17 | 15, 16 | sylibr 235 | . . . 4 ⊢ ((𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → 𝑌:𝑋⟶(𝐴[,)𝐵)) |
18 | 11, 17 | impbii 210 | . . 3 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) ↔ (𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))) |
19 | 18 | a1i 11 | . 2 ⊢ (𝜑 → (𝑌:𝑋⟶(𝐴[,)𝐵) ↔ (𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)))) |
20 | 4, 19 | bitrd 280 | 1 ⊢ (𝜑 → (𝑌 ∈ ((𝐴[,)𝐵) ↑m 𝑋) ↔ (𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ⊆ wss 3933 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 ℝ*cxr 10662 [,)cico 12728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-map 8397 df-xr 10667 df-ico 12732 |
This theorem is referenced by: (None) |
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