Theorem eqeefv 25783

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

Step	Hyp	Ref	Expression
1		eleei 25777	. . 3 |- ( A e. ( EE ` N ) -> A : ( 1 ... N ) --> RR )
2		ffn 6045	. . 3 |- ( A : ( 1 ... N ) --> RR -> A Fn ( 1 ... N ) )
3	1, 2	syl 17	. 2 |- ( A e. ( EE ` N ) -> A Fn ( 1 ... N ) )
4		eleei 25777	. . 3 |- ( B e. ( EE ` N ) -> B : ( 1 ... N ) --> RR )
5		ffn 6045	. . 3 |- ( B : ( 1 ... N ) --> RR -> B Fn ( 1 ... N ) )
6	4, 5	syl 17	. 2 |- ( B e. ( EE ` N ) -> B Fn ( 1 ... N ) )
7		eqfnfv 6311	. 2 |- ( ( A Fn ( 1 ... N ) /\ B Fn ( 1 ... N ) ) -> ( A = B <-> A. i e. ( 1 ... N ) ( A ` i ) = ( B ` i ) ) )
8	3, 6, 7	syl2an 494	1 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( A = B <-> A. i e. ( 1 ... N ) ( A ` i ) = ( B ` i ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   ...cfz 12326   EEcee 25768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ee 25771

This theorem is referenced by:  eqeelen  25784  brbtwn2  25785  colinearalg  25790  axcgrid  25796  ax5seglem4  25812  ax5seglem5  25813  axbtwnid  25819  axeuclid  25843  axcontlem2  25845  axcontlem4  25847  axcontlem7  25850