Theorem 2ffzeq 10461

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 5499	. . . . 5 |- ( F : ( 0 ... M ) --> X -> F Fn ( 0 ... M ) )
2		ffn 5499	. . . . 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 1042	. . 3 |- ( ( M e. NN0 /\ F : ( 0 ... M ) --> X /\ P : ( 0 ... N ) --> Y ) -> ( F Fn ( 0 ... M ) /\ P Fn ( 0 ... N ) ) )
5		eqfnfv2 5767	. . 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 9878	. . . . . . 7 |- ( M e. NN0 <-> M e. ( ZZ>= ` 0 ) )
8		fzopth 10381	. . . . . . 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 6049	. . . . 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 1045	. 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 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   Fn wfn 5338   -->wf 5339   ` cfv 5343  (class class class)co 6041   0cc0 8115   NN0cn0 9484   ZZ>=cuz 9839   ...cfz 10328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314  ax-un 4545  ax-setind 4650  ax-cnex 8206  ax-resscn 8207  ax-1cn 8208  ax-1re 8209  ax-icn 8210  ax-addcl 8211  ax-addrcl 8212  ax-mulcl 8213  ax-addcom 8215  ax-addass 8217  ax-distr 8219  ax-i2m1 8220  ax-0lt1 8221  ax-0id 8223  ax-rnegex 8224  ax-cnre 8226  ax-pre-ltirr 8227  ax-pre-ltwlin 8228  ax-pre-lttrn 8229  ax-pre-apti 8230  ax-pre-ltadd 8231

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-int 3943  df-br 4103  df-opab 4165  df-mpt 4166  df-id 4405  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-ima 4753  df-iota 5303  df-fun 5345  df-fn 5346  df-f 5347  df-fv 5351  df-riota 5994  df-ov 6044  df-oprab 6045  df-mpo 6046  df-pnf 8298  df-mnf 8299  df-xr 8300  df-ltxr 8301  df-le 8302  df-sub 8434  df-neg 8435  df-inn 9226  df-n0 9485  df-z 9564  df-uz 9840  df-fz 10329

This theorem is referenced by:  wlkeq  16319