Theorem 2ffzeq 10333

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 ∈ ℕ0 ∧ 𝐹:(0...𝑀)⟶𝑋 ∧ 𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (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 5472	. . . . 5 ⊢ (𝐹:(0...𝑀)⟶𝑋 → 𝐹 Fn (0...𝑀))
2		ffn 5472	. . . . 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 1039	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐹:(0...𝑀)⟶𝑋 ∧ 𝑃:(0...𝑁)⟶𝑌) → (𝐹 Fn (0...𝑀) ∧ 𝑃 Fn (0...𝑁)))
5		eqfnfv2 5732	. . 3 ⊢ ((𝐹 Fn (0...𝑀) ∧ 𝑃 Fn (0...𝑁)) → (𝐹 = 𝑃 ↔ ((0...𝑀) = (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
6	4, 5	syl 14	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐹:(0...𝑀)⟶𝑋 ∧ 𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ ((0...𝑀) = (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
7		elnn0uz 9756	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (ℤ≥‘0))
8		fzopth 10253	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘0) → ((0...𝑀) = (0...𝑁) ↔ (0 = 0 ∧ 𝑀 = 𝑁)))
9	7, 8	sylbi 121	. . . . . 6 ⊢ (𝑀 ∈ ℕ0 → ((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 ∈ ℕ0 → ((0...𝑀) = (0...𝑁) → 𝑀 = 𝑁))
12	11	anim1d 336	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (((0...𝑀) = (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)) → (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
13		oveq2 6008	. . . . 5 ⊢ (𝑀 = 𝑁 → (0...𝑀) = (0...𝑁))
14	13	anim1i 340	. . . 4 ⊢ ((𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)) → ((0...𝑀) = (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)))
15	12, 14	impbid1 142	. . 3 ⊢ (𝑀 ∈ ℕ0 → (((0...𝑀) = (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
16	15	3ad2ant1 1042	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐹:(0...𝑀)⟶𝑋 ∧ 𝑃:(0...𝑁)⟶𝑌) → (((0...𝑀) = (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
17	6, 16	bitrd 188	1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐹:(0...𝑀)⟶𝑋 ∧ 𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508   Fn wfn 5312  ⟶wf 5313  ‘cfv 5317  (class class class)co 6000  0cc0 7995  ℕ₀cn0 9365  ℤ_≥cuz 9718  ...cfz 10200

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201

This theorem is referenced by: (None)