Theorem 2ffzeq 9613

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

Step	Hyp	Ref	Expression
1		ffn 5174	. . . . 5 |- ( F : ( 0 ... M ) --> X -> F Fn ( 0 ... M ) )
2		ffn 5174	. . . . 5 |- ( P : ( 0 ... N ) --> Y -> P Fn ( 0 ... N ) )
3	1, 2	anim12i 332	. . . 4 |- ( ( F : ( 0 ... M ) --> X /\ P : ( 0 ... N ) --> Y ) -> ( F Fn ( 0 ... M ) /\ P Fn ( 0 ... N ) ) )
4	3	3adant1 962	. . 3 |- ( ( M e. NN0 /\ F : ( 0 ... M ) --> X /\ P : ( 0 ... N ) --> Y ) -> ( F Fn ( 0 ... M ) /\ P Fn ( 0 ... N ) ) )
5		eqfnfv2 5412	. . 3 |- ( ( F Fn ( 0 ... M ) /\ P Fn ( 0 ... N ) ) -> ( F = P <-> ( ( 0 ... M ) = ( 0 ... N ) /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) ) )
6	4, 5	syl 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 9117	. . . . . . 7 |- ( M e. NN0 <-> M e. ( ZZ>= ` 0 ) )
8		fzopth 9536	. . . . . . 7 |- ( M e. ( ZZ>= ` 0 ) -> ( ( 0 ... M ) = ( 0 ... N ) <-> ( 0 = 0 /\ M = N ) ) )
9	7, 8	sylbi 120	. . . . . 6 |- ( M e. NN0 -> ( ( 0 ... M ) = ( 0 ... N ) <-> ( 0 = 0 /\ M = N ) ) )
10		simpr 109	. . . . . 6 |- ( ( 0 = 0 /\ M = N ) -> M = N )
11	9, 10	syl6bi 162	. . . . 5 |- ( M e. NN0 -> ( ( 0 ... M ) = ( 0 ... N ) -> M = N ) )
12	11	anim1d 330	. . . 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 5674	. . . . 5 |- ( M = N -> ( 0 ... M ) = ( 0 ... N ) )
14	13	anim1i 334	. . . 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 ) ) )
15	12, 14	impbid1 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 ) ) ) )
16	15	3ad2ant1 965	. 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 ) ) ) )
17	6, 16	bitrd 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 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 925   = wceq 1290   e. wcel 1439   A.wral 2360   Fn wfn 5023   -->wf 5024   ` cfv 5028  (class class class)co 5666   0cc0 7411   NN0cn0 8734   ZZ>=cuz 9080   ...cfz 9485

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522

This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-inn 8484  df-n0 8735  df-z 8812  df-uz 9081  df-fz 9486

This theorem is referenced by: (None)