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Mirrors > Home > MPE Home > Th. List > Mathboxes > elbigodm | Structured version Visualization version GIF version |
Description: The domain of a function of order G(x) is a subset of the reals. (Contributed by AV, 18-May-2020.) |
Ref | Expression |
---|---|
elbigodm | ⊢ (𝐹 ∈ (Ο‘𝐺) → dom 𝐹 ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elbigo 44618 | . 2 ⊢ (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐺 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)))) | |
2 | reex 10631 | . . . . 5 ⊢ ℝ ∈ V | |
3 | 2, 2 | elpm2 8441 | . . . 4 ⊢ (𝐹 ∈ (ℝ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ)) |
4 | 3 | simprbi 499 | . . 3 ⊢ (𝐹 ∈ (ℝ ↑pm ℝ) → dom 𝐹 ⊆ ℝ) |
5 | 4 | 3ad2ant1 1129 | . 2 ⊢ ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐺 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))) → dom 𝐹 ⊆ ℝ) |
6 | 1, 5 | sylbi 219 | 1 ⊢ (𝐹 ∈ (Ο‘𝐺) → dom 𝐹 ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 ∈ wcel 2113 ∀wral 3141 ∃wrex 3142 ∩ cin 3938 ⊆ wss 3939 class class class wbr 5069 dom cdm 5558 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ↑pm cpm 8410 ℝcr 10539 · cmul 10545 +∞cpnf 10675 ≤ cle 10679 [,)cico 12743 Οcbigo 44614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-pm 8412 df-bigo 44615 |
This theorem is referenced by: (None) |
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