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Theorem 2ffzeq 9517
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 5147 . . . . 5  |-  ( F : ( 0 ... M ) --> X  ->  F  Fn  ( 0 ... M ) )
2 ffn 5147 . . . . 5  |-  ( P : ( 0 ... N ) --> Y  ->  P  Fn  ( 0 ... N ) )
31, 2anim12i 331 . . . 4  |-  ( ( F : ( 0 ... M ) --> X  /\  P : ( 0 ... N ) --> Y )  ->  ( F  Fn  ( 0 ... M )  /\  P  Fn  ( 0 ... N ) ) )
433adant1 961 . . 3  |-  ( ( M  e.  NN0  /\  F : ( 0 ... M ) --> X  /\  P : ( 0 ... N ) --> Y )  ->  ( F  Fn  ( 0 ... M
)  /\  P  Fn  ( 0 ... N
) ) )
5 eqfnfv2 5382 . . 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 9025 . . . . . . 7  |-  ( M  e.  NN0  <->  M  e.  ( ZZ>=
`  0 ) )
8 fzopth 9443 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  0
)  ->  ( (
0 ... M )  =  ( 0 ... N
)  <->  ( 0  =  0  /\  M  =  N ) ) )
97, 8sylbi 119 . . . . . 6  |-  ( M  e.  NN0  ->  ( ( 0 ... M )  =  ( 0 ... N )  <->  ( 0  =  0  /\  M  =  N ) ) )
10 simpr 108 . . . . . 6  |-  ( ( 0  =  0  /\  M  =  N )  ->  M  =  N )
119, 10syl6bi 161 . . . . 5  |-  ( M  e.  NN0  ->  ( ( 0 ... M )  =  ( 0 ... N )  ->  M  =  N ) )
1211anim1d 329 . . . 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 5642 . . . . 5  |-  ( M  =  N  ->  (
0 ... M )  =  ( 0 ... N
) )
1413anim1i 333 . . . 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 140 . . 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 964 . 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 186 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 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   A.wral 2359    Fn wfn 4997   -->wf 4998   ` cfv 5002  (class class class)co 5634   0cc0 7329   NN0cn0 8643   ZZ>=cuz 8988   ...cfz 9393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989  df-fz 9394
This theorem is referenced by: (None)
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