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Theorem eqeefv 26695
Description: Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.)
Assertion
Ref Expression
eqeefv ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴𝑖) = (𝐵𝑖)))
Distinct variable groups:   𝐴,𝑖   𝐵,𝑖   𝑖,𝑁

Proof of Theorem eqeefv
StepHypRef Expression
1 eleei 26689 . . 3 (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ)
21ffnd 6495 . 2 (𝐴 ∈ (𝔼‘𝑁) → 𝐴 Fn (1...𝑁))
3 eleei 26689 . . 3 (𝐵 ∈ (𝔼‘𝑁) → 𝐵:(1...𝑁)⟶ℝ)
43ffnd 6495 . 2 (𝐵 ∈ (𝔼‘𝑁) → 𝐵 Fn (1...𝑁))
5 eqfnfv 6784 . 2 ((𝐴 Fn (1...𝑁) ∧ 𝐵 Fn (1...𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴𝑖) = (𝐵𝑖)))
62, 4, 5syl2an 598 1 ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴𝑖) = (𝐵𝑖)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wral 3130   Fn wfn 6329  cfv 6334  (class class class)co 7140  cr 10525  1c1 10527  ...cfz 12885  𝔼cee 26680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-map 8395  df-ee 26683
This theorem is referenced by:  eqeelen  26696  brbtwn2  26697  colinearalg  26702  axcgrid  26708  ax5seglem4  26724  ax5seglem5  26725  axbtwnid  26731  axeuclid  26755  axcontlem2  26757  axcontlem4  26759  axcontlem7  26762
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