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Theorem eqeefv 24603
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 24597 . . 3  |-  ( A  e.  ( EE `  N )  ->  A : ( 1 ... N ) --> RR )
2 ffn 5405 . . 3  |-  ( A : ( 1 ... N ) --> RR  ->  A  Fn  ( 1 ... N ) )
31, 2syl 15 . 2  |-  ( A  e.  ( EE `  N )  ->  A  Fn  ( 1 ... N
) )
4 eleei 24597 . . 3  |-  ( B  e.  ( EE `  N )  ->  B : ( 1 ... N ) --> RR )
5 ffn 5405 . . 3  |-  ( B : ( 1 ... N ) --> RR  ->  B  Fn  ( 1 ... N ) )
64, 5syl 15 . 2  |-  ( B  e.  ( EE `  N )  ->  B  Fn  ( 1 ... N
) )
7 eqfnfv 5638 . 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 1632    e. wcel 1696   A.wral 2556    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   1c1 8754   ...cfz 10798   EEcee 24588
This theorem is referenced by:  eqeelen  24604  brbtwn2  24605  colinearalg  24610  axcgrid  24616  ax5seglem4  24632  ax5seglem5  24633  axbtwnid  24639  axeuclid  24663  axcontlem2  24665  axcontlem4  24667  axcontlem7  24670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-ee 24591
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