Theorem elicores 46269

Description: Membership in a left-closed, right-open interval with real bounds. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Assertion

Ref	Expression
elicores	⊢ (𝐴 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥[,)𝑦))

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem elicores

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ico 13373	. . . . . 6 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
2	1	reseq1i 5974	. . . . 5 ⊢ ([,) ↾ (ℝ × ℝ)) = ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↾ (ℝ × ℝ))
3		ressxr 11248	. . . . . 6 ⊢ ℝ ⊆ ℝ*
4		resmpo 7530	. . . . . 6 ⊢ ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}))
5	3, 3, 4	mp2an 704	. . . . 5 ⊢ ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
6	2, 5	eqtri 2786	. . . 4 ⊢ ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
7	6	rneqi 5927	. . 3 ⊢ ran ([,) ↾ (ℝ × ℝ)) = ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
8	7	eleq2i 2855	. 2 ⊢ (𝐴 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ 𝐴 ∈ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}))
9		eqid 2763	. . 3 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
10		xrex 13006	. . . 4 ⊢ ℝ* ∈ V
11	10	rabex 5309	. . 3 ⊢ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ V
12	9, 11	elrnmpo 7546	. 2 ⊢ (𝐴 ∈ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
13	3	sseli 3933	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
14	13	adantr 485	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ*)
15	3	sseli 3933	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
16	15	adantl 486	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ*)
17		icoval 13405	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥[,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
18	14, 16, 17	syl2anc 595	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥[,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
19	18	eqcomd 2769	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} = (𝑥[,)𝑦))
20	19	eqeq2d 2774	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐴 = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ↔ 𝐴 = (𝑥[,)𝑦)))
21	20	rexbidva 3187	. . 3 ⊢ (𝑥 ∈ ℝ → (∃𝑦 ∈ ℝ 𝐴 = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ↔ ∃𝑦 ∈ ℝ 𝐴 = (𝑥[,)𝑦)))
22	21	rexbiia 3110	. 2 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥[,)𝑦))
23	8, 12, 22	3bitri 300	1 ⊢ (𝐴 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥[,)𝑦))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416   ⊆ wss 3905   class class class wbr 5109   × cxp 5659  ran crn 5662   ↾ cres 5663  (class class class)co 7410   ∈ cmpo 7412  ℝcr 11094  ℝ^*cxr 11237   < clt 11238   ≤ cle 11239  [,)cico 13369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-xr 11242  df-ico 13373

This theorem is referenced by:  icoresmbl  47277