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Theorem icorempt2 33564
Description: Closed-below, open-above intervals of reals. (Contributed by ML, 26-Jul-2020.)
Hypothesis
Ref Expression
icorempt2.1 𝐹 = ([,) ↾ (ℝ × ℝ))
Assertion
Ref Expression
icorempt2 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem icorempt2
StepHypRef Expression
1 icorempt2.1 . 2 𝐹 = ([,) ↾ (ℝ × ℝ))
2 df-ico 12383 . . . 4 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
32reseq1i 5561 . . 3 ([,) ↾ (ℝ × ℝ)) = ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ))
4 ressxr 10337 . . . 4 ℝ ⊆ ℝ*
5 resmpt2 6956 . . . 4 ((ℝ ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}))
64, 4, 5mp2an 683 . . 3 ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
73, 6eqtri 2787 . 2 ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
8 nfv 2009 . . . 4 𝑧(𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)
9 nfrab1 3270 . . . 4 𝑧{𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}
10 nfrab1 3270 . . . 4 𝑧{𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}
11 rabid 3263 . . . . . . . 8 (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ* ∧ (𝑥𝑧𝑧 < 𝑦)))
12 rexr 10339 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
13 nltmnf 12163 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
1412, 13syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
15 renemnf 10342 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → 𝑥 ≠ -∞)
1615neneqd 2942 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ → ¬ 𝑥 = -∞)
1714, 16jca 507 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → (¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞))
18 pm4.56 1011 . . . . . . . . . . . . . . 15 ((¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞) ↔ ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
1917, 18sylib 209 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
20 mnfxr 10350 . . . . . . . . . . . . . . 15 -∞ ∈ ℝ*
21 xrleloe 12177 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
2212, 20, 21sylancl 580 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
2319, 22mtbird 316 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ¬ 𝑥 ≤ -∞)
24 breq2 4813 . . . . . . . . . . . . . 14 (𝑧 = -∞ → (𝑥𝑧𝑥 ≤ -∞))
2524notbid 309 . . . . . . . . . . . . 13 (𝑧 = -∞ → (¬ 𝑥𝑧 ↔ ¬ 𝑥 ≤ -∞))
2623, 25syl5ibrcom 238 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → (𝑧 = -∞ → ¬ 𝑥𝑧))
2726con2d 131 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (𝑥𝑧 → ¬ 𝑧 = -∞))
28 rexr 10339 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
29 pnfnlt 12162 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
30 breq1 4812 . . . . . . . . . . . . . . 15 (𝑧 = +∞ → (𝑧 < 𝑦 ↔ +∞ < 𝑦))
3130notbid 309 . . . . . . . . . . . . . 14 (𝑧 = +∞ → (¬ 𝑧 < 𝑦 ↔ ¬ +∞ < 𝑦))
3229, 31syl5ibrcom 238 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (𝑧 = +∞ → ¬ 𝑧 < 𝑦))
3332con2d 131 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (𝑧 < 𝑦 → ¬ 𝑧 = +∞))
3428, 33syl 17 . . . . . . . . . . 11 (𝑦 ∈ ℝ → (𝑧 < 𝑦 → ¬ 𝑧 = +∞))
3527, 34im2anan9 613 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥𝑧𝑧 < 𝑦) → (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)))
3635anim2d 605 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ* ∧ (𝑥𝑧𝑧 < 𝑦)) → (𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞))))
37 renepnf 10341 . . . . . . . . . . . 12 (𝑧 ∈ ℝ → 𝑧 ≠ +∞)
3837neneqd 2942 . . . . . . . . . . 11 (𝑧 ∈ ℝ → ¬ 𝑧 = +∞)
3938pm4.71i 555 . . . . . . . . . 10 (𝑧 ∈ ℝ ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
40 xrnemnf 12151 . . . . . . . . . . . 12 ((𝑧 ∈ ℝ*𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ ∨ 𝑧 = +∞))
4140anbi1i 617 . . . . . . . . . . 11 (((𝑧 ∈ ℝ*𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞))
42 df-ne 2938 . . . . . . . . . . . . 13 (𝑧 ≠ -∞ ↔ ¬ 𝑧 = -∞)
4342anbi2i 616 . . . . . . . . . . . 12 ((𝑧 ∈ ℝ*𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞))
4443anbi1i 617 . . . . . . . . . . 11 (((𝑧 ∈ ℝ*𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞))
45 pm5.61 1023 . . . . . . . . . . 11 (((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
4641, 44, 453bitr3i 292 . . . . . . . . . 10 (((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
47 anass 460 . . . . . . . . . 10 (((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)))
4839, 46, 473bitr2ri 291 . . . . . . . . 9 ((𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)) ↔ 𝑧 ∈ ℝ)
4936, 48syl6ib 242 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ* ∧ (𝑥𝑧𝑧 < 𝑦)) → 𝑧 ∈ ℝ))
5011, 49syl5bi 233 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑧 ∈ ℝ))
5111simprbi 490 . . . . . . . 8 (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑥𝑧𝑧 < 𝑦))
5251a1i 11 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑥𝑧𝑧 < 𝑦)))
5350, 52jcad 508 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑧 ∈ ℝ ∧ (𝑥𝑧𝑧 < 𝑦))))
54 rabid 3263 . . . . . 6 (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ ∧ (𝑥𝑧𝑧 < 𝑦)))
5553, 54syl6ibr 243 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}))
56 rabss2 3845 . . . . . . 7 (ℝ ⊆ ℝ* → {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
574, 56ax-mp 5 . . . . . 6 {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ⊆ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}
5857sseli 3757 . . . . 5 (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
5955, 58impbid1 216 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ 𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}))
608, 9, 10, 59eqrd 3780 . . 3 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
6160mpt2eq3ia 6918 . 2 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
621, 7, 613eqtri 2791 1 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  wne 2937  {crab 3059  wss 3732   class class class wbr 4809   × cxp 5275  cres 5279  cmpt2 6844  cr 10188  +∞cpnf 10325  -∞cmnf 10326  *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-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-pre-lttri 10263  ax-pre-lttrn 10264
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  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-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-po 5198  df-so 5199  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-oprab 6846  df-mpt2 6847  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-ico 12383
This theorem is referenced by:  icoreresf  33565  icoreval  33566
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