Theorem 2ffzeq 10548

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 5533	. . . . 5 |- ( F : ( 0 ... M ) --> X -> F Fn ( 0 ... M ) )
2		ffn 5533	. . . . 5 |- ( P : ( 0 ... N ) --> Y -> P Fn ( 0 ... N ) )
3	1, 2	anim12i 338	. . . 4 |- ( ( F : ( 0 ... M ) --> X /\ P : ( 0 ... N ) --> Y ) -> ( F Fn ( 0 ... M ) /\ P Fn ( 0 ... N ) ) )
4	3	3adant1 1046	. . 3 |- ( ( M e. NN0 /\ F : ( 0 ... M ) --> X /\ P : ( 0 ... N ) --> Y ) -> ( F Fn ( 0 ... M ) /\ P Fn ( 0 ... N ) ) )
5		eqfnfv2 5807	. . 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 9960	. . . . . . 7 |- ( M e. NN0 <-> M e. ( ZZ>= ` 0 ) )
8		fzopth 10467	. . . . . . 7 |- ( M e. ( ZZ>= ` 0 ) -> ( ( 0 ... M ) = ( 0 ... N ) <-> ( 0 = 0 /\ M = N ) ) )
9	7, 8	sylbi 121	. . . . . 6 |- ( M e. NN0 -> ( ( 0 ... M ) = ( 0 ... N ) <-> ( 0 = 0 /\ M = N ) ) )
10		simpr 110	. . . . . 6 |- ( ( 0 = 0 /\ M = N ) -> M = N )
11	9, 10	biimtrdi 163	. . . . 5 |- ( M e. NN0 -> ( ( 0 ... M ) = ( 0 ... N ) -> M = N ) )
12	11	anim1d 336	. . . 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 6093	. . . . 5 |- ( M = N -> ( 0 ... M ) = ( 0 ... N ) )
14	13	anim1i 340	. . . 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 142	. . 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 1049	. 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 188	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 ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   Fn wfn 5372   -->wf 5373   ` cfv 5377  (class class class)co 6085   0cc0 8179   NN0cn0 9563   ZZ>=cuz 9921   ...cfz 10411

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412

This theorem is used by:  wlkeq  16595