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Theorem eqeefv 28919
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 28913 . . 3 (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ)
21ffnd 6736 . 2 (𝐴 ∈ (𝔼‘𝑁) → 𝐴 Fn (1...𝑁))
3 eleei 28913 . . 3 (𝐵 ∈ (𝔼‘𝑁) → 𝐵:(1...𝑁)⟶ℝ)
43ffnd 6736 . 2 (𝐵 ∈ (𝔼‘𝑁) → 𝐵 Fn (1...𝑁))
5 eqfnfv 7050 . 2 ((𝐴 Fn (1...𝑁) ∧ 𝐵 Fn (1...𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴𝑖) = (𝐵𝑖)))
62, 4, 5syl2an 596 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 206  wa 395   = wceq 1539  wcel 2107  wral 3060   Fn wfn 6555  cfv 6560  (class class class)co 7432  cr 11155  1c1 11157  ...cfz 13548  𝔼cee 28904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-ee 28907
This theorem is referenced by:  eqeelen  28920  brbtwn2  28921  colinearalg  28926  axcgrid  28932  ax5seglem4  28948  ax5seglem5  28949  axbtwnid  28955  axeuclid  28979  axcontlem2  28981  axcontlem4  28983  axcontlem7  28986
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