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Theorem eqeefv 23294
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 23288 . . 3  |-  ( A  e.  ( EE `  N )  ->  A : ( 1 ... N ) --> RR )
2 ffn 5660 . . 3  |-  ( A : ( 1 ... N ) --> RR  ->  A  Fn  ( 1 ... N ) )
31, 2syl 16 . 2  |-  ( A  e.  ( EE `  N )  ->  A  Fn  ( 1 ... N
) )
4 eleei 23288 . . 3  |-  ( B  e.  ( EE `  N )  ->  B : ( 1 ... N ) --> RR )
5 ffn 5660 . . 3  |-  ( B : ( 1 ... N ) --> RR  ->  B  Fn  ( 1 ... N ) )
64, 5syl 16 . 2  |-  ( B  e.  ( EE `  N )  ->  B  Fn  ( 1 ... N
) )
7 eqfnfv 5899 . 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 1370    e. wcel 1758   A.wral 2795    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193   RRcr 9385   1c1 9387   ...cfz 11547   EEcee 23279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-map 7319  df-ee 23282
This theorem is referenced by:  eqeelen  23295  brbtwn2  23296  colinearalg  23301  axcgrid  23307  ax5seglem4  23323  ax5seglem5  23324  axbtwnid  23330  axeuclid  23354  axcontlem2  23356  axcontlem4  23358  axcontlem7  23361
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