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Theorem elee 28152
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 ∈ β„• β†’ (𝐴 ∈ (π”Όβ€˜π‘) ↔ 𝐴:(1...𝑁)βŸΆβ„))

Proof of Theorem elee
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 oveq2 7417 . . . . 5 (𝑛 = 𝑁 β†’ (1...𝑛) = (1...𝑁))
21oveq2d 7425 . . . 4 (𝑛 = 𝑁 β†’ (ℝ ↑m (1...𝑛)) = (ℝ ↑m (1...𝑁)))
3 df-ee 28149 . . . 4 𝔼 = (𝑛 ∈ β„• ↦ (ℝ ↑m (1...𝑛)))
4 ovex 7442 . . . 4 (ℝ ↑m (1...𝑁)) ∈ V
52, 3, 4fvmpt 6999 . . 3 (𝑁 ∈ β„• β†’ (π”Όβ€˜π‘) = (ℝ ↑m (1...𝑁)))
65eleq2d 2820 . 2 (𝑁 ∈ β„• β†’ (𝐴 ∈ (π”Όβ€˜π‘) ↔ 𝐴 ∈ (ℝ ↑m (1...𝑁))))
7 reex 11201 . . 3 ℝ ∈ V
8 ovex 7442 . . 3 (1...𝑁) ∈ V
97, 8elmap 8865 . 2 (𝐴 ∈ (ℝ ↑m (1...𝑁)) ↔ 𝐴:(1...𝑁)βŸΆβ„)
106, 9bitrdi 287 1 (𝑁 ∈ β„• β†’ (𝐴 ∈ (π”Όβ€˜π‘) ↔ 𝐴:(1...𝑁)βŸΆβ„))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  β„cr 11109  1c1 11111  β„•cn 12212  ...cfz 13484  π”Όcee 28146
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-ee 28149
This theorem is referenced by:  mptelee  28153  eleei  28155  axlowdimlem5  28204  axlowdimlem7  28206  axlowdimlem10  28209  axlowdimlem14  28213  axlowdim1  28217  elntg2  28243
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