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Theorem 2ffzeq 11917
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 5733 . . . . 5  |-  ( F : ( 0 ... M ) --> X  ->  F  Fn  ( 0 ... M ) )
2 ffn 5733 . . . . 5  |-  ( P : ( 0 ... N ) --> Y  ->  P  Fn  ( 0 ... N ) )
31, 2anim12i 570 . . . 4  |-  ( ( F : ( 0 ... M ) --> X  /\  P : ( 0 ... N ) --> Y )  ->  ( F  Fn  ( 0 ... M )  /\  P  Fn  ( 0 ... N ) ) )
433adant1 1027 . . 3  |-  ( ( M  e.  NN0  /\  F : ( 0 ... M ) --> X  /\  P : ( 0 ... N ) --> Y )  ->  ( F  Fn  ( 0 ... M
)  /\  P  Fn  ( 0 ... N
) ) )
5 eqfnfv2 5982 . . 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 17 . 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 11203 . . . . . . 7  |-  ( M  e.  NN0  <->  M  e.  ( ZZ>=
`  0 ) )
8 fzopth 11842 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  0
)  ->  ( (
0 ... M )  =  ( 0 ... N
)  <->  ( 0  =  0  /\  M  =  N ) ) )
97, 8sylbi 199 . . . . . 6  |-  ( M  e.  NN0  ->  ( ( 0 ... M )  =  ( 0 ... N )  <->  ( 0  =  0  /\  M  =  N ) ) )
10 simpr 463 . . . . . 6  |-  ( ( 0  =  0  /\  M  =  N )  ->  M  =  N )
119, 10syl6bi 232 . . . . 5  |-  ( M  e.  NN0  ->  ( ( 0 ... M )  =  ( 0 ... N )  ->  M  =  N ) )
1211anim1d 568 . . . 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 6303 . . . . 5  |-  ( M  =  N  ->  (
0 ... M )  =  ( 0 ... N
) )
1413anim1i 572 . . . 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 207 . . 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 1030 . 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 257 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 setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   A.wral 2739    Fn wfn 5580   -->wf 5581   ` cfv 5585  (class class class)co 6295   0cc0 9544   NN0cn0 10876   ZZ>=cuz 11166   ...cfz 11791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792
This theorem is referenced by:  2wlkeq  25447
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