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Theorem eqeefv 24531
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 24525 . . 3  |-  ( A  e.  ( EE `  N )  ->  A : ( 1 ... N ) --> RR )
2 ffn 5389 . . 3  |-  ( A : ( 1 ... N ) --> RR  ->  A  Fn  ( 1 ... N ) )
31, 2syl 15 . 2  |-  ( A  e.  ( EE `  N )  ->  A  Fn  ( 1 ... N
) )
4 eleei 24525 . . 3  |-  ( B  e.  ( EE `  N )  ->  B : ( 1 ... N ) --> RR )
5 ffn 5389 . . 3  |-  ( B : ( 1 ... N ) --> RR  ->  B  Fn  ( 1 ... N ) )
64, 5syl 15 . 2  |-  ( B  e.  ( EE `  N )  ->  B  Fn  ( 1 ... N
) )
7 eqfnfv 5622 . 2  |-  ( ( A  Fn  ( 1 ... N )  /\  B  Fn  ( 1 ... N ) )  ->  ( A  =  B  <->  A. i  e.  ( 1 ... N ) ( A `  i
)  =  ( B `
 i ) ) )
83, 6, 7syl2an 463 1  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   1c1 8738   ...cfz 10782   EEcee 24516
This theorem is referenced by:  eqeelen  24532  brbtwn2  24533  colinearalg  24538  axcgrid  24544  ax5seglem4  24560  ax5seglem5  24561  axbtwnid  24567  axeuclid  24591  axcontlem2  24593  axcontlem4  24595  axcontlem7  24598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-ee 24519
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