Theorem 2ffzeq 10140

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 5365	. . . . 5 ⊢ (𝐹:(0...𝑀)⟶𝑋 → 𝐹 Fn (0...𝑀))
2		ffn 5365	. . . . 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 1015	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐹:(0...𝑀)⟶𝑋 ∧ 𝑃:(0...𝑁)⟶𝑌) → (𝐹 Fn (0...𝑀) ∧ 𝑃 Fn (0...𝑁)))
5		eqfnfv2 5614	. . 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 9564	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (ℤ≥‘0))
8		fzopth 10060	. . . . . . 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	syl6bi 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 5882	. . . . 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 1018	. 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 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455   Fn wfn 5211  ⟶wf 5212  ‘cfv 5216  (class class class)co 5874  0cc0 7810  ℕ₀cn0 9175  ℤ_≥cuz 9527  ...cfz 10007

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-n0 9176  df-z 9253  df-uz 9528  df-fz 10008

This theorem is referenced by: (None)