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Theorem icorempt2 33431
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 12295 . . . 4 [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
32reseq1i 5499 . . 3 ([,) ↾ (ℝ × ℝ)) = ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ))
4 ressxr 10196 . . . 4 ℝ ⊆ ℝ*
5 resmpt2 6875 . . . 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 710 . . 3 ((𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
73, 6eqtri 2746 . 2 ([,) ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
8 nfv 1956 . . . 4 𝑧(𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)
9 nfrab1 3225 . . . 4 𝑧{𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}
10 nfrab1 3225 . . . 4 𝑧{𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}
11 rabid 3218 . . . . . . . 8 (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ* ∧ (𝑥𝑧𝑧 < 𝑦)))
12 rexr 10198 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
13 nltmnf 12077 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ* → ¬ 𝑥 < -∞)
1412, 13syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ → ¬ 𝑥 < -∞)
15 renemnf 10201 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → 𝑥 ≠ -∞)
1615neneqd 2901 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ → ¬ 𝑥 = -∞)
1714, 16jca 555 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → (¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞))
18 pm4.56 517 . . . . . . . . . . . . . . 15 ((¬ 𝑥 < -∞ ∧ ¬ 𝑥 = -∞) ↔ ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
1917, 18sylib 208 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → ¬ (𝑥 < -∞ ∨ 𝑥 = -∞))
20 mnfxr 10209 . . . . . . . . . . . . . . 15 -∞ ∈ ℝ*
21 xrleloe 12091 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
2212, 20, 21sylancl 697 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (𝑥 ≤ -∞ ↔ (𝑥 < -∞ ∨ 𝑥 = -∞)))
2319, 22mtbird 314 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → ¬ 𝑥 ≤ -∞)
24 breq2 4764 . . . . . . . . . . . . . 14 (𝑧 = -∞ → (𝑥𝑧𝑥 ≤ -∞))
2524notbid 307 . . . . . . . . . . . . 13 (𝑧 = -∞ → (¬ 𝑥𝑧 ↔ ¬ 𝑥 ≤ -∞))
2623, 25syl5ibrcom 237 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → (𝑧 = -∞ → ¬ 𝑥𝑧))
2726con2d 129 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (𝑥𝑧 → ¬ 𝑧 = -∞))
28 rexr 10198 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
29 pnfnlt 12076 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
30 breq1 4763 . . . . . . . . . . . . . . 15 (𝑧 = +∞ → (𝑧 < 𝑦 ↔ +∞ < 𝑦))
3130notbid 307 . . . . . . . . . . . . . 14 (𝑧 = +∞ → (¬ 𝑧 < 𝑦 ↔ ¬ +∞ < 𝑦))
3229, 31syl5ibrcom 237 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (𝑧 = +∞ → ¬ 𝑧 < 𝑦))
3332con2d 129 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (𝑧 < 𝑦 → ¬ 𝑧 = +∞))
3428, 33syl 17 . . . . . . . . . . 11 (𝑦 ∈ ℝ → (𝑧 < 𝑦 → ¬ 𝑧 = +∞))
3527, 34im2anan9 916 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥𝑧𝑧 < 𝑦) → (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)))
3635anim2d 590 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ* ∧ (𝑥𝑧𝑧 < 𝑦)) → (𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞))))
37 renepnf 10200 . . . . . . . . . . . 12 (𝑧 ∈ ℝ → 𝑧 ≠ +∞)
3837neneqd 2901 . . . . . . . . . . 11 (𝑧 ∈ ℝ → ¬ 𝑧 = +∞)
3938pm4.71i 667 . . . . . . . . . 10 (𝑧 ∈ ℝ ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
40 xrnemnf 12065 . . . . . . . . . . . 12 ((𝑧 ∈ ℝ*𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ ∨ 𝑧 = +∞))
4140anbi1i 733 . . . . . . . . . . 11 (((𝑧 ∈ ℝ*𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞))
42 df-ne 2897 . . . . . . . . . . . . 13 (𝑧 ≠ -∞ ↔ ¬ 𝑧 = -∞)
4342anbi2i 732 . . . . . . . . . . . 12 ((𝑧 ∈ ℝ*𝑧 ≠ -∞) ↔ (𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞))
4443anbi1i 733 . . . . . . . . . . 11 (((𝑧 ∈ ℝ*𝑧 ≠ -∞) ∧ ¬ 𝑧 = +∞) ↔ ((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞))
45 pm5.61 751 . . . . . . . . . . 11 (((𝑧 ∈ ℝ ∨ 𝑧 = +∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
4641, 44, 453bitr3i 290 . . . . . . . . . 10 (((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ ∧ ¬ 𝑧 = +∞))
47 anass 684 . . . . . . . . . 10 (((𝑧 ∈ ℝ* ∧ ¬ 𝑧 = -∞) ∧ ¬ 𝑧 = +∞) ↔ (𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)))
4839, 46, 473bitr2ri 289 . . . . . . . . 9 ((𝑧 ∈ ℝ* ∧ (¬ 𝑧 = -∞ ∧ ¬ 𝑧 = +∞)) ↔ 𝑧 ∈ ℝ)
4936, 48syl6ib 241 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ* ∧ (𝑥𝑧𝑧 < 𝑦)) → 𝑧 ∈ ℝ))
5011, 49syl5bi 232 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑧 ∈ ℝ))
5111simprbi 483 . . . . . . . 8 (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑥𝑧𝑧 < 𝑦))
5251a1i 11 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑥𝑧𝑧 < 𝑦)))
5350, 52jcad 556 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → (𝑧 ∈ ℝ ∧ (𝑥𝑧𝑧 < 𝑦))))
54 rabid 3218 . . . . . 6 (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ (𝑧 ∈ ℝ ∧ (𝑥𝑧𝑧 < 𝑦)))
5553, 54syl6ibr 242 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}))
56 rabss2 3791 . . . . . . 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 3705 . . . . 5 (𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)} → 𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
5955, 58impbid1 215 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} ↔ 𝑧 ∈ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)}))
608, 9, 10, 59eqrd 3728 . . 3 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)} = {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
6160mpt2eq3ia 6837 . 2 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
621, 7, 613eqtri 2750 1 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1596  wcel 2103  wne 2896  {crab 3018  wss 3680   class class class wbr 4760   × cxp 5216  cres 5220  cmpt2 6767  cr 10048  +∞cpnf 10184  -∞cmnf 10185  *cxr 10186   < clt 10187  cle 10188  [,)cico 12291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-pre-lttri 10123  ax-pre-lttrn 10124
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-po 5139  df-so 5140  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-oprab 6769  df-mpt2 6770  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-ico 12295
This theorem is referenced by:  icoreresf  33432  icoreval  33433
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