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Theorem eqeefv 25583
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 25577 . . 3 (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ)
2 ffn 5958 . . 3 (𝐴:(1...𝑁)⟶ℝ → 𝐴 Fn (1...𝑁))
31, 2syl 17 . 2 (𝐴 ∈ (𝔼‘𝑁) → 𝐴 Fn (1...𝑁))
4 eleei 25577 . . 3 (𝐵 ∈ (𝔼‘𝑁) → 𝐵:(1...𝑁)⟶ℝ)
5 ffn 5958 . . 3 (𝐵:(1...𝑁)⟶ℝ → 𝐵 Fn (1...𝑁))
64, 5syl 17 . 2 (𝐵 ∈ (𝔼‘𝑁) → 𝐵 Fn (1...𝑁))
7 eqfnfv 6219 . 2 ((𝐴 Fn (1...𝑁) ∧ 𝐵 Fn (1...𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴𝑖) = (𝐵𝑖)))
83, 6, 7syl2an 493 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 195  wa 383   = wceq 1475  wcel 1977  wral 2896   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  cr 9814  1c1 9816  ...cfz 12197  𝔼cee 25568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-ee 25571
This theorem is referenced by:  eqeelen  25584  brbtwn2  25585  colinearalg  25590  axcgrid  25596  ax5seglem4  25612  ax5seglem5  25613  axbtwnid  25619  axeuclid  25643  axcontlem2  25645  axcontlem4  25647  axcontlem7  25650
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