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Theorem pfxval 40679
Description: Value of a prefix. (Contributed by AV, 2-May-2020.)
Assertion
Ref Expression
pfxval ((𝑆𝑉𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))

Proof of Theorem pfxval
Dummy variables 𝑙 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pfx 40678 . . 3 prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))
21a1i 11 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩)))
3 simpl 473 . . . 4 ((𝑠 = 𝑆𝑙 = 𝐿) → 𝑠 = 𝑆)
4 opeq2 4371 . . . . 5 (𝑙 = 𝐿 → ⟨0, 𝑙⟩ = ⟨0, 𝐿⟩)
54adantl 482 . . . 4 ((𝑠 = 𝑆𝑙 = 𝐿) → ⟨0, 𝑙⟩ = ⟨0, 𝐿⟩)
63, 5oveq12d 6622 . . 3 ((𝑠 = 𝑆𝑙 = 𝐿) → (𝑠 substr ⟨0, 𝑙⟩) = (𝑆 substr ⟨0, 𝐿⟩))
76adantl 482 . 2 (((𝑆𝑉𝐿 ∈ ℕ0) ∧ (𝑠 = 𝑆𝑙 = 𝐿)) → (𝑠 substr ⟨0, 𝑙⟩) = (𝑆 substr ⟨0, 𝐿⟩))
8 elex 3198 . . 3 (𝑆𝑉𝑆 ∈ V)
98adantr 481 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → 𝑆 ∈ V)
10 simpr 477 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → 𝐿 ∈ ℕ0)
11 ovex 6632 . . 3 (𝑆 substr ⟨0, 𝐿⟩) ∈ V
1211a1i 11 . 2 ((𝑆𝑉𝐿 ∈ ℕ0) → (𝑆 substr ⟨0, 𝐿⟩) ∈ V)
132, 7, 9, 10, 12ovmpt2d 6741 1 ((𝑆𝑉𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cop 4154  (class class class)co 6604  cmpt2 6606  0cc0 9880  0cn0 11236   substr csubstr 13234   prefix cpfx 40677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-pfx 40678
This theorem is referenced by:  pfx00  40680  pfx0  40681  pfxcl  40682  pfxmpt  40683  pfxid  40688  pfxn0  40690  pfxnd  40691  pfxfv  40695  pfx1  40707  pfxswrd  40709  swrdpfx  40710  pfxpfx  40711  pfxccatpfx1  40723  pfxccatpfx2  40724  splvalpfx  40731  cshword2  40733  pfxco  40734
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