Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  icoreresf Structured version   Visualization version   GIF version

Theorem icoreresf 36962
Description: Closed-below, open-above intervals of reals map to subsets of reals. (Contributed by ML, 25-Jul-2020.)
Assertion
Ref Expression
icoreresf ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ

Proof of Theorem icoreresf
Dummy variables 𝑥 𝑦 𝑧 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexpssxrxp 11291 . . 3 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
2 df-ico 13365 . . . . 5 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
32ixxf 13369 . . . 4 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
4 ffn 6723 . . . 4 ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → [,) Fn (ℝ* × ℝ*))
5 fnssresb 6678 . . . 4 ([,) Fn (ℝ* × ℝ*) → (([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) ↔ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)))
63, 4, 5mp2b 10 . . 3 (([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) ↔ (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
71, 6mpbir 230 . 2 ([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ)
8 eqid 2725 . . . . 5 ([,) ↾ (ℝ × ℝ)) = ([,) ↾ (ℝ × ℝ))
98icorempo 36961 . . . 4 ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
109rneqi 5939 . . 3 ran ([,) ↾ (ℝ × ℝ)) = ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
11 ssrab2 4073 . . . . . 6 {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ
12 reex 11231 . . . . . . 7 ℝ ∈ V
1312elpw2 5348 . . . . . 6 ({𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ ↔ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ)
1411, 13mpbir 230 . . . . 5 {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ
1514rgen2w 3055 . . . 4 𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ
16 eqid 2725 . . . . . . . 8 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
1716rnmpo 7554 . . . . . . 7 ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) = {𝑙 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}}
1817eqabri 2869 . . . . . 6 (𝑙 ∈ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
19 simpl 481 . . . . . . . . 9 ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ)
20 simpr 483 . . . . . . . . 9 ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
2119, 20r19.29d2r 3129 . . . . . . . 8 ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ({𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}))
22 eleq1 2813 . . . . . . . . . . 11 (𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑙 ∈ 𝒫 ℝ ↔ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ))
2322biimparc 478 . . . . . . . . . 10 (({𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ)
2423a1i 11 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (({𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ))
2524rexlimivv 3189 . . . . . . . 8 (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ({𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ)
2621, 25syl 17 . . . . . . 7 ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ)
2726ex 411 . . . . . 6 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑙 ∈ 𝒫 ℝ))
2818, 27biimtrid 241 . . . . 5 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ → (𝑙 ∈ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ))
2928ssrdv 3982 . . . 4 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ → ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) ⊆ 𝒫 ℝ)
3015, 29ax-mp 5 . . 3 ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) ⊆ 𝒫 ℝ
3110, 30eqsstri 4011 . 2 ran ([,) ↾ (ℝ × ℝ)) ⊆ 𝒫 ℝ
32 df-f 6553 . 2 (([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ ↔ (([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) ∧ ran ([,) ↾ (ℝ × ℝ)) ⊆ 𝒫 ℝ))
337, 31, 32mpbir2an 709 1 ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059  {crab 3418  wss 3944  𝒫 cpw 4604   class class class wbr 5149   × cxp 5676  ran crn 5679  cres 5680   Fn wfn 6544  wf 6545  cmpo 7421  cr 11139  *cxr 11279   < clt 11280  cle 11281  [,)cico 13361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-pre-lttri 11214  ax-pre-lttrn 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-ico 13365
This theorem is referenced by:  icoreelrnab  36964  icoreunrn  36969
  Copyright terms: Public domain W3C validator