elbigof - Mathbox for Alexander van der Vekens

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Theorem elbigof 45334

Description: A function of order G(x) is a function. (Contributed by AV, 18-May-2020.)

**Assertion**
Ref	Expression
elbigof	⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ)

Proof of Theorem elbigof Dummy variables 𝑥 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elbigo 45331	. 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 10667	. . . . 5 ⊢ ℝ ∈ V
3	2, 2	elpm2 8457	. . . 4 ⊢ (𝐹 ∈ (ℝ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ))
4	3	simplbi 502	. . 3 ⊢ (𝐹 ∈ (ℝ ↑_pm ℝ) → 𝐹:dom 𝐹⟶ℝ)
5	4	3ad2ant1 1131	. 2 ⊢ ((𝐹 ∈ (ℝ ↑_pm ℝ) ∧ 𝐺 ∈ (ℝ ↑_pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))) → 𝐹:dom 𝐹⟶ℝ)
6	1, 5	sylbi 220	1 ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1085 ∈ wcel 2112 ∀wral 3071 ∃wrex 3072 ∩ cin 3858 ⊆ wss 3859 class class class wbr 5033 dom cdm 5525 ⟶wf 6332 ‘cfv 6336 (class class class)co 7151 ↑_pm cpm 8418 ℝcr 10575 · cmul 10581 +∞cpnf 10711 ≤ cle 10715 [,)cico 12782 Οcbigo 45327

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5170 ax-nul 5177 ax-pow 5235 ax-pr 5299 ax-un 7460 ax-cnex 10632 ax-resscn 10633

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-ral 3076 df-rex 3077 df-rab 3080 df-v 3412 df-sbc 3698 df-dif 3862 df-un 3864 df-in 3866 df-ss 3876 df-nul 4227 df-if 4422 df-pw 4497 df-sn 4524 df-pr 4526 df-op 4530 df-uni 4800 df-br 5034 df-opab 5096 df-mpt 5114 df-id 5431 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-fv 6344 df-ov 7154 df-oprab 7155 df-mpo 7156 df-pm 8420 df-bigo 45328

This theorem is referenced by: (None)

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