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 | ovex 6577 | . . . 4 ⊢ (𝐴[,)𝐵) ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴[,)𝐵) ∈ V) |
3 | elhoi.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | elmapg 7757 | . . 3 ⊢ (((𝐴[,)𝐵) ∈ V ∧ 𝑋 ∈ 𝑉) → (𝑌 ∈ ((𝐴[,)𝐵) ↑𝑚 𝑋) ↔ 𝑌:𝑋⟶(𝐴[,)𝐵))) | |
5 | 2, 3, 4 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝑌 ∈ ((𝐴[,)𝐵) ↑𝑚 𝑋) ↔ 𝑌:𝑋⟶(𝐴[,)𝐵))) |
6 | id 22 | . . . . . 6 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → 𝑌:𝑋⟶(𝐴[,)𝐵)) | |
7 | icossxr 12129 | . . . . . . 7 ⊢ (𝐴[,)𝐵) ⊆ ℝ* | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → (𝐴[,)𝐵) ⊆ ℝ*) |
9 | 6, 8 | fssd 5970 | . . . . 5 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → 𝑌:𝑋⟶ℝ*) |
10 | ffvelrn 6265 | . . . . . 6 ⊢ ((𝑌:𝑋⟶(𝐴[,)𝐵) ∧ 𝑥 ∈ 𝑋) → (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) | |
11 | 10 | ralrimiva 2949 | . . . . 5 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) |
12 | 9, 11 | jca 553 | . . . 4 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → (𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))) |
13 | ffn 5958 | . . . . . . 7 ⊢ (𝑌:𝑋⟶ℝ* → 𝑌 Fn 𝑋) | |
14 | 13 | adantr 480 | . . . . . 6 ⊢ ((𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → 𝑌 Fn 𝑋) |
15 | simpr 476 | . . . . . 6 ⊢ ((𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) | |
16 | 14, 15 | jca 553 | . . . . 5 ⊢ ((𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → (𝑌 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))) |
17 | ffnfv 6295 | . . . . 5 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) ↔ (𝑌 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))) | |
18 | 16, 17 | sylibr 223 | . . . 4 ⊢ ((𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → 𝑌:𝑋⟶(𝐴[,)𝐵)) |
19 | 12, 18 | impbii 198 | . . 3 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) ↔ (𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))) |
20 | 19 | a1i 11 | . 2 ⊢ (𝜑 → (𝑌:𝑋⟶(𝐴[,)𝐵) ↔ (𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)))) |
21 | 5, 20 | bitrd 267 | 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 195 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ℝ*cxr 9952 [,)cico 12048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-xr 9957 df-ico 12052 |
This theorem is referenced by: (None) |
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