elhoi - Mathbox for Glauco Siliprandi

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Theorem elhoi 47077

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 7426	. . 3 ⊢ (𝜑 → (𝐴[,)𝐵) ∈ V)
2		elhoi.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
3		elmapg 8814	. . 3 ⊢ (((𝐴[,)𝐵) ∈ V ∧ 𝑋 ∈ 𝑉) → (𝑌 ∈ ((𝐴[,)𝐵) ↑_m 𝑋) ↔ 𝑌:𝑋⟶(𝐴[,)𝐵)))
4	1, 2, 3	syl2anc 593	. 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 13430	. . . . . . 7 ⊢ (𝐴[,)𝐵) ⊆ ℝ^*
7	6	a1i 11	. . . . . 6 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → (𝐴[,)𝐵) ⊆ ℝ^*)
8	5, 7	fssd 6704	. . . . 5 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → 𝑌:𝑋⟶ℝ^*)
9		ffvelcdm 7057	. . . . . 6 ⊢ ((𝑌:𝑋⟶(𝐴[,)𝐵) ∧ 𝑥 ∈ 𝑋) → (𝑌‘𝑥) ∈ (𝐴[,)𝐵))
10	9	ralrimiva 3153	. . . . 5 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))
11	8, 10	jca 519	. . . 4 ⊢ (𝑌:𝑋⟶(𝐴[,)𝐵) → (𝑌:𝑋⟶ℝ^* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)))
12		ffn 6686	. . . . . . 7 ⊢ (𝑌:𝑋⟶ℝ^* → 𝑌 Fn 𝑋)
13	12	adantr 484	. . . . . 6 ⊢ ((𝑌:𝑋⟶ℝ^* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → 𝑌 Fn 𝑋)
14		simpr 488	. . . . . 6 ⊢ ((𝑌:𝑋⟶ℝ^* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵))
15	13, 14	jca 519	. . . . 5 ⊢ ((𝑌:𝑋⟶ℝ^* ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)) → (𝑌 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑌‘𝑥) ∈ (𝐴[,)𝐵)))
16		ffnfv 7095	. . . . 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 399 ∈ wcel 2141 ∀wral 3075 Vcvv 3453 ⊆ wss 3902 Fn wfn 6511 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 ↑_m cmap 8802 ℝ^*cxr 11209 [,)cico 13345

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-fv 6524 df-ov 7394 df-oprab 7395 df-mpo 7396 df-1st 7965 df-2nd 7966 df-map 8804 df-xr 11214 df-ico 13349

This theorem is referenced by: (None)

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