Theorem elee 28692

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.

Step	Hyp	Ref	Expression
1		oveq2 7422	. . . . 5 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
2	1	oveq2d 7430	. . . 4 ⊢ (𝑛 = 𝑁 → (ℝ ↑m (1...𝑛)) = (ℝ ↑m (1...𝑁)))
3		df-ee 28689	. . . 4 ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛)))
4		ovex 7447	. . . 4 ⊢ (ℝ ↑m (1...𝑁)) ∈ V
5	2, 3, 4	fvmpt 6999	. . 3 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (ℝ ↑m (1...𝑁)))
6	5	eleq2d 2814	. 2 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (ℝ ↑m (1...𝑁))))
7		reex 11221	. . 3 ⊢ ℝ ∈ V
8		ovex 7447	. . 3 ⊢ (1...𝑁) ∈ V
9	7, 8	elmap 8881	. 2 ⊢ (𝐴 ∈ (ℝ ↑m (1...𝑁)) ↔ 𝐴:(1...𝑁)⟶ℝ)
10	6, 9	bitrdi 287	1 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1534   ∈ wcel 2099  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414   ↑_m cmap 8836  ℝcr 11129  1c1 11131  ℕcn 12234  ...cfz 13508  𝔼cee 28686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8838  df-ee 28689

This theorem is referenced by:  mptelee  28693  eleei  28695  axlowdimlem5  28744  axlowdimlem7  28746  axlowdimlem10  28749  axlowdimlem14  28753  axlowdim1  28757  elntg2  28783