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

Theorem frec2uz0d 9694
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 0 or 1 for the upper integers ), 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 5253 . 2 (𝐺‘∅) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘∅)
3 frec2uz.1 . . 3 (𝜑𝐶 ∈ ℤ)
4 frec0g 6093 . . 3 (𝐶 ∈ ℤ → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘∅) = 𝐶)
53, 4syl 14 . 2 (𝜑 → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘∅) = 𝐶)
62, 5syl5eq 2127 1 (𝜑 → (𝐺‘∅) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  c0 3269  cmpt 3865  cfv 4968  (class class class)co 5590  freccfrec 6086  1c1 7253   + caddc 7255  cz 8645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368  df-xp 4406  df-rel 4407  df-cnv 4408  df-co 4409  df-dm 4410  df-res 4412  df-iota 4933  df-fun 4970  df-fn 4971  df-fv 4976  df-recs 6001  df-frec 6087
This theorem is referenced by:  frec2uzuzd  9697  frec2uzrand  9700  frec2uzrdg  9704  frecuzrdgg  9711  frecfzennn  9721  0tonninf  9733  1tonninf  9734  omgadd  10044
  Copyright terms: Public domain W3C validator