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Theorem eqeefv 25717
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 25711 . . 3 (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ)
2 ffn 6012 . . 3 (𝐴:(1...𝑁)⟶ℝ → 𝐴 Fn (1...𝑁))
31, 2syl 17 . 2 (𝐴 ∈ (𝔼‘𝑁) → 𝐴 Fn (1...𝑁))
4 eleei 25711 . . 3 (𝐵 ∈ (𝔼‘𝑁) → 𝐵:(1...𝑁)⟶ℝ)
5 ffn 6012 . . 3 (𝐵:(1...𝑁)⟶ℝ → 𝐵 Fn (1...𝑁))
64, 5syl 17 . 2 (𝐵 ∈ (𝔼‘𝑁) → 𝐵 Fn (1...𝑁))
7 eqfnfv 6277 . 2 ((𝐴 Fn (1...𝑁) ∧ 𝐵 Fn (1...𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴𝑖) = (𝐵𝑖)))
83, 6, 7syl2an 494 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 196  wa 384   = wceq 1480  wcel 1987  wral 2908   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  cr 9895  1c1 9897  ...cfz 12284  𝔼cee 25702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-ee 25705
This theorem is referenced by:  eqeelen  25718  brbtwn2  25719  colinearalg  25724  axcgrid  25730  ax5seglem4  25746  ax5seglem5  25747  axbtwnid  25753  axeuclid  25777  axcontlem2  25779  axcontlem4  25781  axcontlem7  25784
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