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Theorem ello1mpt 14459
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)
2 eqid 2770 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
31, 2fmptd 6527 . . 3 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
4 ello1mpt.1 . . 3 (𝜑𝐴 ⊆ ℝ)
5 ello12 14454 . . 3 (((𝑥𝐴𝐵):𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚)))
63, 4, 5syl2anc 565 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚)))
7 nfv 1994 . . . . . 6 𝑥 𝑦𝑧
8 nffvmpt1 6340 . . . . . . 7 𝑥((𝑥𝐴𝐵)‘𝑧)
9 nfcv 2912 . . . . . . 7 𝑥
10 nfcv 2912 . . . . . . 7 𝑥𝑚
118, 9, 10nfbr 4831 . . . . . 6 𝑥((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚
127, 11nfim 1976 . . . . 5 𝑥(𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚)
13 nfv 1994 . . . . 5 𝑧(𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚)
14 breq2 4788 . . . . . 6 (𝑧 = 𝑥 → (𝑦𝑧𝑦𝑥))
15 fveq2 6332 . . . . . . 7 (𝑧 = 𝑥 → ((𝑥𝐴𝐵)‘𝑧) = ((𝑥𝐴𝐵)‘𝑥))
1615breq1d 4794 . . . . . 6 (𝑧 = 𝑥 → (((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚 ↔ ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚))
1714, 16imbi12d 333 . . . . 5 (𝑧 = 𝑥 → ((𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚) ↔ (𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚)))
1812, 13, 17cbvral 3315 . . . 4 (∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥𝐴 (𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚))
19 simpr 471 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥𝐴)
202fvmpt2 6433 . . . . . . . 8 ((𝑥𝐴𝐵 ∈ ℝ) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2119, 1, 20syl2anc 565 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2221breq1d 4794 . . . . . 6 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚𝐵𝑚))
2322imbi2d 329 . . . . 5 ((𝜑𝑥𝐴) → ((𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚) ↔ (𝑦𝑥𝐵𝑚)))
2423ralbidva 3133 . . . 4 (𝜑 → (∀𝑥𝐴 (𝑦𝑥 → ((𝑥𝐴𝐵)‘𝑥) ≤ 𝑚) ↔ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))
2518, 24syl5bb 272 . . 3 (𝜑 → (∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))
26252rexbidv 3204 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → ((𝑥𝐴𝐵)‘𝑧) ≤ 𝑚) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))
276, 26bitrd 268 1 (𝜑 → ((𝑥𝐴𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥𝐴 (𝑦𝑥𝐵𝑚)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wral 3060  wrex 3061  wss 3721   class class class wbr 4784  cmpt 4861  wf 6027  cfv 6031  cr 10136  cle 10276  ≤𝑂(1)clo1 14425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-pre-lttri 10211  ax-pre-lttrn 10212
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-er 7895  df-pm 8011  df-en 8109  df-dom 8110  df-sdom 8111  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-ico 12385  df-lo1 14429
This theorem is referenced by:  ello1mpt2  14460  ello1d  14461  elo1mpt  14472  o1lo1  14475  lo1resb  14502  lo1add  14564  lo1mul  14565  lo1le  14589
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