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Theorem frec2uz0d 10418
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 5531 . 2 (𝐺‘∅) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘∅)
3 frec2uz.1 . . 3 (𝜑𝐶 ∈ ℤ)
4 frec0g 6416 . . 3 (𝐶 ∈ ℤ → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘∅) = 𝐶)
53, 4syl 14 . 2 (𝜑 → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘∅) = 𝐶)
62, 5eqtrid 2234 1 (𝜑 → (𝐺‘∅) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  c0 3437  cmpt 4079  cfv 5231  (class class class)co 5891  freccfrec 6409  1c1 7831   + caddc 7833  cz 9272
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-res 4653  df-iota 5193  df-fun 5233  df-fn 5234  df-fv 5239  df-recs 6324  df-frec 6410
This theorem is referenced by:  frec2uzuzd  10421  frec2uzrand  10424  frec2uzrdg  10428  frecuzrdgg  10435  frecfzennn  10445  0tonninf  10458  1tonninf  10459  omgadd  10801  ennnfonelem1  12432  ennnfonelemhf1o  12438  012of  15149  2o01f  15150  isomninnlem  15182  iswomninnlem  15201  ismkvnnlem  15204
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