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Theorem ello1mpt 14939
 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 6876 . . 3 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
3 ello1mpt.1 . . 3 (𝜑𝐴 ⊆ ℝ)
4 ello12 14934 . . 3 (((𝑥𝐴𝐵):𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚)))
52, 3, 4syl2anc 587 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚)))
6 nfv 1915 . . . . . 6 𝑥 𝑦𝑧
7 nffvmpt1 6674 . . . . . . 7 𝑥((𝑥𝐴𝐵)‘𝑧)
8 nfcv 2919 . . . . . . 7 𝑥
9 nfcv 2919 . . . . . . 7 𝑥𝑚
107, 8, 9nfbr 5083 . . . . . 6 𝑥((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚
116, 10nfim 1897 . . . . 5 𝑥(𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚)
12 nfv 1915 . . . . 5 𝑧(𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚)
13 breq2 5040 . . . . . 6 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
14 fveq2 6663 . . . . . . 7 (𝑧 = 𝑥 → ((𝑥𝐴𝐵)‘𝑧) = ((𝑥𝐴𝐵)‘𝑥))
1514breq1d 5046 . . . . . 6 (𝑧 = 𝑥 → (((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚 ↔ ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚))
1613, 15imbi12d 348 . . . . 5 (𝑧 = 𝑥 → ((𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚) ↔ (𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚)))
1711, 12, 16cbvralw 3352 . . . 4 (∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥𝐴 (𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚))
18 simpr 488 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥𝐴)
19 eqid 2758 . . . . . . . . 9 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
2019fvmpt2 6775 . . . . . . . 8 ((𝑥𝐴𝐵 ∈ ℝ) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2118, 1, 20syl2anc 587 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2221breq1d 5046 . . . . . 6 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚𝐵𝑚))
2322imbi2d 344 . . . . 5 ((𝜑𝑥𝐴) → ((𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚) ↔ (𝑦𝑥𝐵𝑚)))
2423ralbidva 3125 . . . 4 (𝜑 → (∀𝑥𝐴 (𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚) ↔ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))
2517, 24syl5bb 286 . . 3 (𝜑 → (∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))
26252rexbidv 3224 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))
275, 26bitrd 282 1 (𝜑 → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071   ⊆ wss 3860   class class class wbr 5036   ↦ cmpt 5116  ⟶wf 6336  ‘cfv 6340  ℝcr 10587   ≤ cle 10727  ≤𝑂(1)clo1 14905 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-pre-lttri 10662  ax-pre-lttrn 10663 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-po 5447  df-so 5448  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-er 8305  df-pm 8425  df-en 8541  df-dom 8542  df-sdom 8543  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-ico 12798  df-lo1 14909 This theorem is referenced by:  ello1mpt2  14940  ello1d  14941  elo1mpt  14952  o1lo1  14955  lo1resb  14982  lo1add  15044  lo1mul  15045  lo1le  15069
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