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Theorem eqeefv 23100
Description: Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.)
Assertion
Ref Expression
eqeefv  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
Distinct variable groups:    A, i    B, i    i, N

Proof of Theorem eqeefv
StepHypRef Expression
1 eleei 23094 . . 3  |-  ( A  e.  ( EE `  N )  ->  A : ( 1 ... N ) --> RR )
2 ffn 5554 . . 3  |-  ( A : ( 1 ... N ) --> RR  ->  A  Fn  ( 1 ... N ) )
31, 2syl 16 . 2  |-  ( A  e.  ( EE `  N )  ->  A  Fn  ( 1 ... N
) )
4 eleei 23094 . . 3  |-  ( B  e.  ( EE `  N )  ->  B : ( 1 ... N ) --> RR )
5 ffn 5554 . . 3  |-  ( B : ( 1 ... N ) --> RR  ->  B  Fn  ( 1 ... N ) )
64, 5syl 16 . 2  |-  ( B  e.  ( EE `  N )  ->  B  Fn  ( 1 ... N
) )
7 eqfnfv 5792 . 2  |-  ( ( A  Fn  ( 1 ... N )  /\  B  Fn  ( 1 ... N ) )  ->  ( A  =  B  <->  A. i  e.  ( 1 ... N ) ( A `  i
)  =  ( B `
 i ) ) )
83, 6, 7syl2an 477 1  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   RRcr 9273   1c1 9275   ...cfz 11429   EEcee 23085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-map 7208  df-ee 23088
This theorem is referenced by:  eqeelen  23101  brbtwn2  23102  colinearalg  23107  axcgrid  23113  ax5seglem4  23129  ax5seglem5  23130  axbtwnid  23136  axeuclid  23160  axcontlem2  23162  axcontlem4  23164  axcontlem7  23167
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