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Theorem frec2uz0d 10416
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 5530 . 2 (𝐺‘∅) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘∅)
3 frec2uz.1 . . 3 (𝜑𝐶 ∈ ℤ)
4 frec0g 6415 . . 3 (𝐶 ∈ ℤ → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘∅) = 𝐶)
53, 4syl 14 . 2 (𝜑 → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘∅) = 𝐶)
62, 5eqtrid 2233 1 (𝜑 → (𝐺‘∅) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wcel 2159  c0 3436  cmpt 4078  cfv 5230  (class class class)co 5890  freccfrec 6408  1c1 7829   + caddc 7831  cz 9270
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-iord 4380  df-on 4382  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-res 4652  df-iota 5192  df-fun 5232  df-fn 5233  df-fv 5238  df-recs 6323  df-frec 6409
This theorem is referenced by:  frec2uzuzd  10419  frec2uzrand  10422  frec2uzrdg  10426  frecuzrdgg  10433  frecfzennn  10443  0tonninf  10456  1tonninf  10457  omgadd  10799  ennnfonelem1  12425  ennnfonelemhf1o  12431  012of  15129  2o01f  15130  isomninnlem  15162  iswomninnlem  15181  ismkvnnlem  15184
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