Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  frec2uz0d GIF version

Theorem frec2uz0d 8866
 Description: The mapping 𝐺 is a one-to-one mapping from 𝜔 onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number 𝐶 (normally 0 for the upper integers ℕ0 or 1 for the upper integers ℕ), 1 maps to 𝐶 + 1, etc. This theorem shows the value of 𝐺 at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.)
Hypotheses
Ref Expression
frec2uz.1 (φ𝐶 ℤ)
frec2uz.2 𝐺 = frec((x ℤ ↦ (x + 1)), 𝐶)
Assertion
Ref Expression
frec2uz0d (φ → (𝐺‘∅) = 𝐶)
Distinct variable group:   x,𝐶
Allowed substitution hints:   φ(x)   𝐺(x)

Proof of Theorem frec2uz0d
StepHypRef Expression
1 frec2uz.2 . . 3 𝐺 = frec((x ℤ ↦ (x + 1)), 𝐶)
21fveq1i 5122 . 2 (𝐺‘∅) = (frec((x ℤ ↦ (x + 1)), 𝐶)‘∅)
3 frec2uz.1 . . 3 (φ𝐶 ℤ)
4 frec0g 5922 . . 3 (𝐶 ℤ → (frec((x ℤ ↦ (x + 1)), 𝐶)‘∅) = 𝐶)
53, 4syl 14 . 2 (φ → (frec((x ℤ ↦ (x + 1)), 𝐶)‘∅) = 𝐶)
62, 5syl5eq 2081 1 (φ → (𝐺‘∅) = 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  ∅c0 3218   ↦ cmpt 3809  ‘cfv 4845  (class class class)co 5455  freccfrec 5917  1c1 6712   + caddc 6714  ℤcz 8021 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-recs 5861  df-frec 5918 This theorem is referenced by:  frec2uzzd  8867  frec2uzuzd  8869  frec2uzrand  8872  frec2uzrdg  8876  frecfzennn  8884
 Copyright terms: Public domain W3C validator