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Theorem icoreresf 35873
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 11208 . . 3 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
2 df-ico 13279 . . . . 5 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
32ixxf 13283 . . . 4 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
4 ffn 6672 . . . 4 ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → [,) Fn (ℝ* × ℝ*))
5 fnssresb 6627 . . . 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 2733 . . . . 5 ([,) ↾ (ℝ × ℝ)) = ([,) ↾ (ℝ × ℝ))
98icorempo 35872 . . . 4 ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
109rneqi 5896 . . 3 ran ([,) ↾ (ℝ × ℝ)) = ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
11 ssrab2 4041 . . . . . 6 {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ
12 reex 11150 . . . . . . 7 ℝ ∈ V
1312elpw2 5306 . . . . . 6 ({𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ ↔ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ ℝ)
1411, 13mpbir 230 . . . . 5 {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ
1514rgen2w 3066 . . . 4 𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ
16 eqid 2733 . . . . . . . 8 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
1716rnmpo 7493 . . . . . . 7 ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) = {𝑙 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}}
1817eqabi 2878 . . . . . 6 (𝑙 ∈ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
19 simpl 484 . . . . . . . . 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 486 . . . . . . . . 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 3134 . . . . . . . 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 2822 . . . . . . . . . . 11 (𝑙 = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑙 ∈ 𝒫 ℝ ↔ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ∈ 𝒫 ℝ))
2322biimparc 481 . . . . . . . . . 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 3193 . . . . . . . 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 414 . . . . . 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 3954 . . . 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 3982 . 2 ran ([,) ↾ (ℝ × ℝ)) ⊆ 𝒫 ℝ
32 df-f 6504 . 2 (([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ ↔ (([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) ∧ ran ([,) ↾ (ℝ × ℝ)) ⊆ 𝒫 ℝ))
337, 31, 32mpbir2an 710 1 ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  wrex 3070  {crab 3406  wss 3914  𝒫 cpw 4564   class class class wbr 5109   × cxp 5635  ran crn 5638  cres 5639   Fn wfn 6495  wf 6496  cmpo 7363  cr 11058  *cxr 11196   < clt 11197  cle 11198  [,)cico 13275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-pre-lttri 11133  ax-pre-lttrn 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-po 5549  df-so 5550  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-ico 13279
This theorem is referenced by:  icoreelrnab  35875  icoreunrn  35880
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