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 7193 | . . 3 ⊢ (𝜑 → (𝐴[,)𝐵) ∈ V) | |
2 | elhoi.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | elmapg 8421 | . . 3 ⊢ (((𝐴[,)𝐵) ∈ V ∧ 𝑋 ∈ 𝑉) → (𝑌 ∈ ((𝐴[,)𝐵) ↑m 𝑋) ↔ 𝑌:𝑋⟶(𝐴[,)𝐵))) | |
4 | 1, 2, 3 | syl2anc 586 | . 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 12824 | . . . . . . 7 ⊢ (𝐴[,)𝐵) ⊆ ℝ* | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → (𝐴[,)𝐵) ⊆ ℝ*) |
8 | 5, 7 | fssd 6530 | . . . . 5 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → 𝑌:𝑋⟶ℝ*) |
9 | ffvelrn 6851 | . . . . . 6 ⊢ ((𝑌:𝑋⟶(𝐴[,)𝐵) ∧ 𝑥 ∈ 𝑋) → (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) | |
10 | 9 | ralrimiva 3184 | . . . . 5 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) |
11 | 8, 10 | jca 514 | . . . 4 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → (𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))) |
12 | ffn 6516 | . . . . . . 7 ⊢ (𝑌:𝑋⟶ℝ* → 𝑌 Fn 𝑋) | |
13 | 12 | adantr 483 | . . . . . 6 ⊢ ((𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → 𝑌 Fn 𝑋) |
14 | simpr 487 | . . . . . 6 ⊢ ((𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) | |
15 | 13, 14 | jca 514 | . . . . 5 ⊢ ((𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → (𝑌 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))) |
16 | ffnfv 6884 | . . . . 5 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) ↔ (𝑌 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))) | |
17 | 15, 16 | sylibr 236 | . . . 4 ⊢ ((𝑌:𝑋⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → 𝑌:𝑋⟶(𝐴[,)𝐵)) |
18 | 11, 17 | impbii 211 | . . 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 281 | 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 208 ∧ wa 398 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ⊆ wss 3938 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 ℝ*cxr 10676 [,)cico 12743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-xr 10681 df-ico 12747 |
This theorem is referenced by: (None) |
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