elbigof - Mathbox for Alexander van der Vekens

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Theorem elbigof 49137

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 49134	. 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 11158	. . . . 5 ⊢ ℝ ∈ V
3	2, 2	elpm2 8850	. . . 4 ⊢ (𝐹 ∈ (ℝ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ))
4	3	simplbi 500	. . 3 ⊢ (𝐹 ∈ (ℝ ↑_pm ℝ) → 𝐹:dom 𝐹⟶ℝ)
5	4	3ad2ant1 1145	. 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 219	1 ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1097 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 ∩ cin 3901 ⊆ wss 3902 class class class wbr 5097 dom cdm 5643 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 ↑_pm cpm 8803 ℝcr 11066 · cmul 11072 +∞cpnf 11207 ≤ cle 11211 [,)cico 13345 Οcbigo 49130

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3743 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-fv 6524 df-ov 7394 df-oprab 7395 df-mpo 7396 df-pm 8805 df-bigo 49131

This theorem is referenced by: (None)

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