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Theorem 2ffzeq 12654
 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 ∈ ℕ0𝐹:(0...𝑀)⟶𝑋𝑃:(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 6206 . . . . 5 (𝐹:(0...𝑀)⟶𝑋𝐹 Fn (0...𝑀))
2 ffn 6206 . . . . 5 (𝑃:(0...𝑁)⟶𝑌𝑃 Fn (0...𝑁))
31, 2anim12i 591 . . . 4 ((𝐹:(0...𝑀)⟶𝑋𝑃:(0...𝑁)⟶𝑌) → (𝐹 Fn (0...𝑀) ∧ 𝑃 Fn (0...𝑁)))
433adant1 1125 . . 3 ((𝑀 ∈ ℕ0𝐹:(0...𝑀)⟶𝑋𝑃:(0...𝑁)⟶𝑌) → (𝐹 Fn (0...𝑀) ∧ 𝑃 Fn (0...𝑁)))
5 eqfnfv2 6475 . . 3 ((𝐹 Fn (0...𝑀) ∧ 𝑃 Fn (0...𝑁)) → (𝐹 = 𝑃 ↔ ((0...𝑀) = (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
64, 5syl 17 . 2 ((𝑀 ∈ ℕ0𝐹:(0...𝑀)⟶𝑋𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ ((0...𝑀) = (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
7 elnn0uz 11918 . . . . . . 7 (𝑀 ∈ ℕ0𝑀 ∈ (ℤ‘0))
8 fzopth 12571 . . . . . . 7 (𝑀 ∈ (ℤ‘0) → ((0...𝑀) = (0...𝑁) ↔ (0 = 0 ∧ 𝑀 = 𝑁)))
97, 8sylbi 207 . . . . . 6 (𝑀 ∈ ℕ0 → ((0...𝑀) = (0...𝑁) ↔ (0 = 0 ∧ 𝑀 = 𝑁)))
10 simpr 479 . . . . . 6 ((0 = 0 ∧ 𝑀 = 𝑁) → 𝑀 = 𝑁)
119, 10syl6bi 243 . . . . 5 (𝑀 ∈ ℕ0 → ((0...𝑀) = (0...𝑁) → 𝑀 = 𝑁))
1211anim1d 589 . . . 4 (𝑀 ∈ ℕ0 → (((0...𝑀) = (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖)) → (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
13 oveq2 6821 . . . . 5 (𝑀 = 𝑁 → (0...𝑀) = (0...𝑁))
1413anim1i 593 . . . 4 ((𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖)) → ((0...𝑀) = (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖)))
1512, 14impbid1 215 . . 3 (𝑀 ∈ ℕ0 → (((0...𝑀) = (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
16153ad2ant1 1128 . 2 ((𝑀 ∈ ℕ0𝐹:(0...𝑀)⟶𝑋𝑃:(0...𝑁)⟶𝑌) → (((0...𝑀) = (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
176, 16bitrd 268 1 ((𝑀 ∈ ℕ0𝐹:(0...𝑀)⟶𝑋𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050   Fn wfn 6044  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813  0cc0 10128  ℕ0cn0 11484  ℤ≥cuz 11879  ...cfz 12519 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520 This theorem is referenced by:  wlkeq  26739
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