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Theorem 2ffzeq 9925
Description: Two functions over 0 based finite set of sequential integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
Assertion
Ref Expression
2ffzeq  |-  ( ( M  e.  NN0  /\  F : ( 0 ... M ) --> X  /\  P : ( 0 ... N ) --> Y )  ->  ( F  =  P  <->  ( M  =  N  /\  A. i  e.  ( 0 ... M
) ( F `  i )  =  ( P `  i ) ) ) )
Distinct variable groups:    i, F    i, M    P, i
Allowed substitution hints:    N( i)    X( i)    Y( i)

Proof of Theorem 2ffzeq
StepHypRef Expression
1 ffn 5272 . . . . 5  |-  ( F : ( 0 ... M ) --> X  ->  F  Fn  ( 0 ... M ) )
2 ffn 5272 . . . . 5  |-  ( P : ( 0 ... N ) --> Y  ->  P  Fn  ( 0 ... N ) )
31, 2anim12i 336 . . . 4  |-  ( ( F : ( 0 ... M ) --> X  /\  P : ( 0 ... N ) --> Y )  ->  ( F  Fn  ( 0 ... M )  /\  P  Fn  ( 0 ... N ) ) )
433adant1 999 . . 3  |-  ( ( M  e.  NN0  /\  F : ( 0 ... M ) --> X  /\  P : ( 0 ... N ) --> Y )  ->  ( F  Fn  ( 0 ... M
)  /\  P  Fn  ( 0 ... N
) ) )
5 eqfnfv2 5519 . . 3  |-  ( ( F  Fn  ( 0 ... M )  /\  P  Fn  ( 0 ... N ) )  ->  ( F  =  P  <->  ( ( 0 ... M )  =  ( 0 ... N
)  /\  A. i  e.  ( 0 ... M
) ( F `  i )  =  ( P `  i ) ) ) )
64, 5syl 14 . 2  |-  ( ( M  e.  NN0  /\  F : ( 0 ... M ) --> X  /\  P : ( 0 ... N ) --> Y )  ->  ( F  =  P  <->  ( ( 0 ... M )  =  ( 0 ... N
)  /\  A. i  e.  ( 0 ... M
) ( F `  i )  =  ( P `  i ) ) ) )
7 elnn0uz 9370 . . . . . . 7  |-  ( M  e.  NN0  <->  M  e.  ( ZZ>=
`  0 ) )
8 fzopth 9848 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  0
)  ->  ( (
0 ... M )  =  ( 0 ... N
)  <->  ( 0  =  0  /\  M  =  N ) ) )
97, 8sylbi 120 . . . . . 6  |-  ( M  e.  NN0  ->  ( ( 0 ... M )  =  ( 0 ... N )  <->  ( 0  =  0  /\  M  =  N ) ) )
10 simpr 109 . . . . . 6  |-  ( ( 0  =  0  /\  M  =  N )  ->  M  =  N )
119, 10syl6bi 162 . . . . 5  |-  ( M  e.  NN0  ->  ( ( 0 ... M )  =  ( 0 ... N )  ->  M  =  N ) )
1211anim1d 334 . . . 4  |-  ( M  e.  NN0  ->  ( ( ( 0 ... M
)  =  ( 0 ... N )  /\  A. i  e.  ( 0 ... M ) ( F `  i )  =  ( P `  i ) )  -> 
( M  =  N  /\  A. i  e.  ( 0 ... M
) ( F `  i )  =  ( P `  i ) ) ) )
13 oveq2 5782 . . . . 5  |-  ( M  =  N  ->  (
0 ... M )  =  ( 0 ... N
) )
1413anim1i 338 . . . 4  |-  ( ( M  =  N  /\  A. i  e.  ( 0 ... M ) ( F `  i )  =  ( P `  i ) )  -> 
( ( 0 ... M )  =  ( 0 ... N )  /\  A. i  e.  ( 0 ... M
) ( F `  i )  =  ( P `  i ) ) )
1512, 14impbid1 141 . . 3  |-  ( M  e.  NN0  ->  ( ( ( 0 ... M
)  =  ( 0 ... N )  /\  A. i  e.  ( 0 ... M ) ( F `  i )  =  ( P `  i ) )  <->  ( M  =  N  /\  A. i  e.  ( 0 ... M
) ( F `  i )  =  ( P `  i ) ) ) )
16153ad2ant1 1002 . 2  |-  ( ( M  e.  NN0  /\  F : ( 0 ... M ) --> X  /\  P : ( 0 ... N ) --> Y )  ->  ( ( ( 0 ... M )  =  ( 0 ... N )  /\  A. i  e.  ( 0 ... M ) ( F `  i )  =  ( P `  i ) )  <->  ( M  =  N  /\  A. i  e.  ( 0 ... M
) ( F `  i )  =  ( P `  i ) ) ) )
176, 16bitrd 187 1  |-  ( ( M  e.  NN0  /\  F : ( 0 ... M ) --> X  /\  P : ( 0 ... N ) --> Y )  ->  ( F  =  P  <->  ( M  =  N  /\  A. i  e.  ( 0 ... M
) ( F `  i )  =  ( P `  i ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   A.wral 2416    Fn wfn 5118   -->wf 5119   ` cfv 5123  (class class class)co 5774   0cc0 7627   NN0cn0 8984   ZZ>=cuz 9333   ...cfz 9797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-fz 9798
This theorem is referenced by: (None)
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