Theorem eqeefv 26683

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)(𝐴‘𝑖) = (𝐵‘𝑖)))

Distinct variable groups:   𝐴,𝑖   𝐵,𝑖   𝑖,𝑁

Proof of Theorem eqeefv

Step	Hyp	Ref	Expression
1		eleei 26677	. . 3 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ)
2	1	ffnd 6509	. 2 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴 Fn (1...𝑁))
3		eleei 26677	. . 3 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝐵:(1...𝑁)⟶ℝ)
4	3	ffnd 6509	. 2 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝐵 Fn (1...𝑁))
5		eqfnfv 6796	. 2 ⊢ ((𝐴 Fn (1...𝑁) ∧ 𝐵 Fn (1...𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (𝐵‘𝑖)))
6	2, 4, 5	syl2an 597	1 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (𝐵‘𝑖)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138   Fn wfn 6344  ‘cfv 6349  (class class class)co 7150  ℝcr 10530  1c1 10532  ...cfz 12886  𝔼cee 26668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8402  df-ee 26671

This theorem is referenced by:  eqeelen  26684  brbtwn2  26685  colinearalg  26690  axcgrid  26696  ax5seglem4  26712  ax5seglem5  26713  axbtwnid  26719  axeuclid  26743  axcontlem2  26745  axcontlem4  26747  axcontlem7  26750