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Theorem ello1mpt 14866
Description: Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
Hypotheses
Ref Expression
ello1mpt.1 (𝜑𝐴 ⊆ ℝ)
ello1mpt.2 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
Assertion
Ref Expression
ello1mpt (𝜑 → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))
Distinct variable groups:   𝑥,𝑚,𝑦,𝐴   𝐵,𝑚,𝑦   𝜑,𝑚,𝑥,𝑦
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem ello1mpt
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ello1mpt.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
21fmpttd 6871 . . 3 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
3 ello1mpt.1 . . 3 (𝜑𝐴 ⊆ ℝ)
4 ello12 14861 . . 3 (((𝑥𝐴𝐵):𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚)))
52, 3, 4syl2anc 584 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚)))
6 nfv 1906 . . . . . 6 𝑥 𝑦𝑧
7 nffvmpt1 6674 . . . . . . 7 𝑥((𝑥𝐴𝐵)‘𝑧)
8 nfcv 2974 . . . . . . 7 𝑥
9 nfcv 2974 . . . . . . 7 𝑥𝑚
107, 8, 9nfbr 5104 . . . . . 6 𝑥((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚
116, 10nfim 1888 . . . . 5 𝑥(𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚)
12 nfv 1906 . . . . 5 𝑧(𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚)
13 breq2 5061 . . . . . 6 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
14 fveq2 6663 . . . . . . 7 (𝑧 = 𝑥 → ((𝑥𝐴𝐵)‘𝑧) = ((𝑥𝐴𝐵)‘𝑥))
1514breq1d 5067 . . . . . 6 (𝑧 = 𝑥 → (((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚 ↔ ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚))
1613, 15imbi12d 346 . . . . 5 (𝑧 = 𝑥 → ((𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚) ↔ (𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚)))
1711, 12, 16cbvralw 3439 . . . 4 (∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥𝐴 (𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚))
18 simpr 485 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥𝐴)
19 eqid 2818 . . . . . . . . 9 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
2019fvmpt2 6771 . . . . . . . 8 ((𝑥𝐴𝐵 ∈ ℝ) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2118, 1, 20syl2anc 584 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2221breq1d 5067 . . . . . 6 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚𝐵𝑚))
2322imbi2d 342 . . . . 5 ((𝜑𝑥𝐴) → ((𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚) ↔ (𝑦𝑥𝐵𝑚)))
2423ralbidva 3193 . . . 4 (𝜑 → (∀𝑥𝐴 (𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚) ↔ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))
2517, 24syl5bb 284 . . 3 (𝜑 → (∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))
26252rexbidv 3297 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))
275, 26bitrd 280 1 (𝜑 → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933   class class class wbr 5057  cmpt 5137  wf 6344  cfv 6348  cr 10524  cle 10664  ≤𝑂(1)clo1 14832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-pre-lttri 10599  ax-pre-lttrn 10600
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-er 8278  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-ico 12732  df-lo1 14836
This theorem is referenced by:  ello1mpt2  14867  ello1d  14868  elo1mpt  14879  o1lo1  14882  lo1resb  14909  lo1add  14971  lo1mul  14972  lo1le  14996
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