elhoi - Mathbox for Glauco Siliprandi

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Theorem elhoi 44080

Description: Membership in a multidimensional half-open interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypothesis**
Ref	Expression
elhoi.1	⊢ (𝜑 → 𝑋 ∈ 𝑉)

**Assertion**
Ref	Expression
elhoi	⊢ (𝜑 → (𝑌 ∈ ((𝐴[,)𝐵) ↑_m 𝑋) ↔ (𝑌:𝑋⟶ℝ^* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝑋 𝑥,𝑌 Allowed substitution hints: 𝜑(𝑥) 𝑉(𝑥)

**Proof of Theorem elhoi**
Step	Hyp	Ref	Expression
1		ovexd 7310	. . 3 ⊢ (𝜑 → (𝐴[,)𝐵) ∈ V)
2		elhoi.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
3		elmapg 8628	. . 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 13164	. . . . . . 7 ⊢ (𝐴[,)𝐵) ⊆ ℝ^*
7	6	a1i 11	. . . . . 6 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → (𝐴[,)𝐵) ⊆ ℝ^*)
8	5, 7	fssd 6618	. . . . 5 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → 𝑌:𝑋⟶ℝ^*)
9		ffvelrn 6959	. . . . . 6 ⊢ ((𝑌:𝑋⟶(𝐴[,)𝐵) ∧ 𝑥 ∈ 𝑋) → (𝑌‘𝑥) ∈ (𝐴[,)𝐵))
10	9	ralrimiva 3103	. . . . 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 6600	. . . . . . 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 6992	. . . . 5 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) ↔ (𝑌 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)))
17	15, 16	sylibr 233	. . . 4 ⊢ ((𝑌:𝑋⟶ℝ^* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → 𝑌:𝑋⟶(𝐴[,)𝐵))
18	11, 17	impbii 208	. . 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 278	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 205 ∧ wa 396 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ⊆ wss 3887 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 ℝ^*cxr 11008 [,)cico 13081

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-map 8617 df-xr 11013 df-ico 13085

This theorem is referenced by: (None)

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