Theorem 2ffzeq 10376

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 5482	. . . . 5 |- ( F : ( 0 ... M ) --> X -> F Fn ( 0 ... M ) )
2		ffn 5482	. . . . 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 1041	. . 3 |- ( ( M e. NN0 /\ F : ( 0 ... M ) --> X /\ P : ( 0 ... N ) --> Y ) -> ( F Fn ( 0 ... M ) /\ P Fn ( 0 ... N ) ) )
5		eqfnfv2 5745	. . 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 9794	. . . . . . 7 |- ( M e. NN0 <-> M e. ( ZZ>= ` 0 ) )
8		fzopth 10296	. . . . . . 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 6026	. . . . 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 1044	. 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 1004   = wceq 1397   e. wcel 2202   A.wral 2510   Fn wfn 5321   -->wf 5322   ` cfv 5326  (class class class)co 6018   0cc0 8032   NN0cn0 9402   ZZ>=cuz 9755   ...cfz 10243

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244

This theorem is referenced by:  wlkeq  16207