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Theorem icoreresf 32853
 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 10031 . . 3 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
2 df-ico 12126 . . . . 5 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
32ixxf 12130 . . . 4 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
4 ffn 6004 . . . 4 ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → [,) Fn (ℝ* × ℝ*))
5 fnssresb 5963 . . . 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 221 . 2 ([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ)
8 eqid 2621 . . . . 5 ([,) ↾ (ℝ × ℝ)) = ([,) ↾ (ℝ × ℝ))
98icorempt2 32852 . . . 4 ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
109rneqi 5314 . . 3 ran ([,) ↾ (ℝ × ℝ)) = ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
11 ssrab2 3668 . . . . . 6 {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ
12 reex 9974 . . . . . . 7 ℝ ∈ V
1312elpw2 4790 . . . . . 6 ({𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ ↔ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ)
1411, 13mpbir 221 . . . . 5 {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ
1514rgen2w 2920 . . . 4 𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ
16 eqid 2621 . . . . . . . 8 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
1716rnmpt2 6726 . . . . . . 7 ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) = {𝑙 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}}
1817abeq2i 2732 . . . . . 6 (𝑙 ∈ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
19 simpl 473 . . . . . . . . 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 477 . . . . . . . . 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 3072 . . . . . . . 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 2686 . . . . . . . . . . 11 (𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑙 ∈ 𝒫 ℝ ↔ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ))
2322biimparc 504 . . . . . . . . . 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 3029 . . . . . . . 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 450 . . . . . 6 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑙 ∈ 𝒫 ℝ))
2818, 27syl5bi 232 . . . . 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 3590 . . . 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 3616 . 2 ran ([,) ↾ (ℝ × ℝ)) ⊆ 𝒫 ℝ
32 df-f 5853 . 2 (([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ ↔ (([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) ∧ ran ([,) ↾ (ℝ × ℝ)) ⊆ 𝒫 ℝ))
337, 31, 32mpbir2an 954 1 ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  {crab 2911   ⊆ wss 3556  𝒫 cpw 4132   class class class wbr 4615   × cxp 5074  ran crn 5077   ↾ cres 5078   Fn wfn 5844  ⟶wf 5845   ↦ cmpt2 6609  ℝcr 9882  ℝ*cxr 10020   < clt 10021   ≤ cle 10022  [,)cico 12122 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-pre-lttri 9957  ax-pre-lttrn 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-po 4997  df-so 4998  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-ico 12126 This theorem is referenced by:  icoreelrnab  32855  icoreunrn  32860
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