Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  2ffzeq GIF version

Theorem 2ffzeq 8768
 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 0 𝐹:(0...𝑀)⟶𝑋 𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 𝑖 (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
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 4989 . . . . 5 (𝐹:(0...𝑀)⟶𝑋𝐹 Fn (0...𝑀))
2 ffn 4989 . . . . 5 (𝑃:(0...𝑁)⟶𝑌𝑃 Fn (0...𝑁))
31, 2anim12i 321 . . . 4 ((𝐹:(0...𝑀)⟶𝑋 𝑃:(0...𝑁)⟶𝑌) → (𝐹 Fn (0...𝑀) 𝑃 Fn (0...𝑁)))
433adant1 921 . . 3 ((𝑀 0 𝐹:(0...𝑀)⟶𝑋 𝑃:(0...𝑁)⟶𝑌) → (𝐹 Fn (0...𝑀) 𝑃 Fn (0...𝑁)))
5 eqfnfv2 5209 . . 3 ((𝐹 Fn (0...𝑀) 𝑃 Fn (0...𝑁)) → (𝐹 = 𝑃 ↔ ((0...𝑀) = (0...𝑁) 𝑖 (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
64, 5syl 14 . 2 ((𝑀 0 𝐹:(0...𝑀)⟶𝑋 𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ ((0...𝑀) = (0...𝑁) 𝑖 (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
7 elnn0uz 8286 . . . . . . 7 (𝑀 0𝑀 (ℤ‘0))
8 fzopth 8694 . . . . . . 7 (𝑀 (ℤ‘0) → ((0...𝑀) = (0...𝑁) ↔ (0 = 0 𝑀 = 𝑁)))
97, 8sylbi 114 . . . . . 6 (𝑀 0 → ((0...𝑀) = (0...𝑁) ↔ (0 = 0 𝑀 = 𝑁)))
10 simpr 103 . . . . . 6 ((0 = 0 𝑀 = 𝑁) → 𝑀 = 𝑁)
119, 10syl6bi 152 . . . . 5 (𝑀 0 → ((0...𝑀) = (0...𝑁) → 𝑀 = 𝑁))
1211anim1d 319 . . . 4 (𝑀 0 → (((0...𝑀) = (0...𝑁) 𝑖 (0...𝑀)(𝐹𝑖) = (𝑃𝑖)) → (𝑀 = 𝑁 𝑖 (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
13 oveq2 5463 . . . . 5 (𝑀 = 𝑁 → (0...𝑀) = (0...𝑁))
1413anim1i 323 . . . 4 ((𝑀 = 𝑁 𝑖 (0...𝑀)(𝐹𝑖) = (𝑃𝑖)) → ((0...𝑀) = (0...𝑁) 𝑖 (0...𝑀)(𝐹𝑖) = (𝑃𝑖)))
1512, 14impbid1 130 . . 3 (𝑀 0 → (((0...𝑀) = (0...𝑁) 𝑖 (0...𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 𝑖 (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
16153ad2ant1 924 . 2 ((𝑀 0 𝐹:(0...𝑀)⟶𝑋 𝑃:(0...𝑁)⟶𝑌) → (((0...𝑀) = (0...𝑁) 𝑖 (0...𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 𝑖 (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
176, 16bitrd 177 1 ((𝑀 0 𝐹:(0...𝑀)⟶𝑋 𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 𝑖 (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∀wral 2300   Fn wfn 4840  ⟶wf 4841  ‘cfv 4845  (class class class)co 5455  0cc0 6711  ℕ0cn0 7957  ℤ≥cuz 8249  ...cfz 8644 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-addcom 6783  ax-addass 6785  ax-distr 6787  ax-i2m1 6788  ax-0id 6791  ax-rnegex 6792  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-ltwlin 6796  ax-pre-lttrn 6797  ax-pre-apti 6798  ax-pre-ltadd 6799 This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863  df-sub 6981  df-neg 6982  df-inn 7696  df-n0 7958  df-z 8022  df-uz 8250  df-fz 8645 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator