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Theorem elicores 40330
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.
StepHypRef Expression
1 df-ico 12383 . . . . . 6 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
21reseq1i 5561 . . . . 5 ([,) ↾ (ℝ × ℝ)) = ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ))
3 ressxr 10337 . . . . . 6 ℝ ⊆ ℝ*
4 resmpt2 6956 . . . . . 6 ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}))
53, 3, 4mp2an 683 . . . . 5 ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
62, 5eqtri 2787 . . . 4 ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
76rneqi 5520 . . 3 ran ([,) ↾ (ℝ × ℝ)) = ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
87eleq2i 2836 . 2 (𝐴 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ 𝐴 ∈ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}))
9 eqid 2765 . . 3 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
10 xrex 12025 . . . 4 * ∈ V
1110rabex 4973 . . 3 {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ V
129, 11elrnmpt2 6971 . 2 (𝐴 ∈ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
133sseli 3757 . . . . . . . 8 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
1413adantr 472 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ*)
153sseli 3757 . . . . . . . 8 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
1615adantl 473 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ*)
17 icoval 12415 . . . . . . 7 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥[,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
1814, 16, 17syl2anc 579 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥[,)𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
1918eqcomd 2771 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} = (𝑥[,)𝑦))
2019eqeq2d 2775 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐴 = {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ 𝐴 = (𝑥[,)𝑦)))
2120rexbidva 3196 . . 3 (𝑥 ∈ ℝ → (∃𝑦 ∈ ℝ 𝐴 = {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ ∃𝑦 ∈ ℝ 𝐴 = (𝑥[,)𝑦)))
2221rexbiia 3187 . 2 (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥[,)𝑦))
238, 12, 223bitri 288 1 (𝐴 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥[,)𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155  wrex 3056  {crab 3059  wss 3732   class class class wbr 4809   × cxp 5275  ran crn 5278  cres 5279  (class class class)co 6842  cmpt2 6844  cr 10188  *cxr 10327   < clt 10328  cle 10329  [,)cico 12379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-xr 10332  df-ico 12383
This theorem is referenced by:  icoresmbl  41329
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