Theorem 2ffzeq 10097

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 5347	. . . . 5 ⊢ (𝐹:(0...𝑀)⟶𝑋 → 𝐹 Fn (0...𝑀))
2		ffn 5347	. . . . 5 ⊢ (𝑃:(0...𝑁)⟶𝑌 → 𝑃 Fn (0...𝑁))
3	1, 2	anim12i 336	. . . 4 ⊢ ((𝐹:(0...𝑀)⟶𝑋 ∧ 𝑃:(0...𝑁)⟶𝑌) → (𝐹 Fn (0...𝑀) ∧ 𝑃 Fn (0...𝑁)))
4	3	3adant1 1010	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐹:(0...𝑀)⟶𝑋 ∧ 𝑃:(0...𝑁)⟶𝑌) → (𝐹 Fn (0...𝑀) ∧ 𝑃 Fn (0...𝑁)))
5		eqfnfv2 5594	. . 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 9524	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (ℤ≥‘0))
8		fzopth 10017	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘0) → ((0...𝑀) = (0...𝑁) ↔ (0 = 0 ∧ 𝑀 = 𝑁)))
9	7, 8	sylbi 120	. . . . . 6 ⊢ (𝑀 ∈ ℕ0 → ((0...𝑀) = (0...𝑁) ↔ (0 = 0 ∧ 𝑀 = 𝑁)))
10		simpr 109	. . . . . 6 ⊢ ((0 = 0 ∧ 𝑀 = 𝑁) → 𝑀 = 𝑁)
11	9, 10	syl6bi 162	. . . . 5 ⊢ (𝑀 ∈ ℕ0 → ((0...𝑀) = (0...𝑁) → 𝑀 = 𝑁))
12	11	anim1d 334	. . . 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 5861	. . . . 5 ⊢ (𝑀 = 𝑁 → (0...𝑀) = (0...𝑁))
14	13	anim1i 338	. . . 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 141	. . 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 1013	. 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 187	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 103   ↔ wb 104   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  ∀wral 2448   Fn wfn 5193  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  0cc0 7774  ℕ₀cn0 9135  ℤ_≥cuz 9487  ...cfz 9965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966

This theorem is referenced by: (None)