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Theorem elee 25973
 Description: Membership in a Euclidean space. We define Euclidean space here using Cartesian coordinates over 𝑁 space. We later abstract away from this using Tarski's geometry axioms, so this exact definition is unimportant. (Contributed by Scott Fenton, 3-Jun-2013.)
Assertion
Ref Expression
elee (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ))

Proof of Theorem elee
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 oveq2 6821 . . . . 5 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21oveq2d 6829 . . . 4 (𝑛 = 𝑁 → (ℝ ↑𝑚 (1...𝑛)) = (ℝ ↑𝑚 (1...𝑁)))
3 df-ee 25970 . . . 4 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑𝑚 (1...𝑛)))
4 ovex 6841 . . . 4 (ℝ ↑𝑚 (1...𝑁)) ∈ V
52, 3, 4fvmpt 6444 . . 3 (𝑁 ∈ ℕ → (𝔼‘𝑁) = (ℝ ↑𝑚 (1...𝑁)))
65eleq2d 2825 . 2 (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (ℝ ↑𝑚 (1...𝑁))))
7 reex 10219 . . 3 ℝ ∈ V
8 ovex 6841 . . 3 (1...𝑁) ∈ V
97, 8elmap 8052 . 2 (𝐴 ∈ (ℝ ↑𝑚 (1...𝑁)) ↔ 𝐴:(1...𝑁)⟶ℝ)
106, 9syl6bb 276 1 (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1632   ∈ wcel 2139  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813   ↑𝑚 cmap 8023  ℝcr 10127  1c1 10129  ℕcn 11212  ...cfz 12519  𝔼cee 25967 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-ee 25970 This theorem is referenced by:  mptelee  25974  eleei  25976  axlowdimlem5  26025  axlowdimlem7  26027  axlowdimlem10  26030  axlowdimlem14  26034  axlowdim1  26038
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