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Theorem tfr1on 6201
Description: Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 12-Mar-2022.)
Hypotheses
Ref Expression
tfr1on.f 𝐹 = recs(𝐺)
tfr1on.g (𝜑 → Fun 𝐺)
tfr1on.x (𝜑 → Ord 𝑋)
tfr1on.ex ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
tfr1on.u ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
tfr1on.yx (𝜑𝑌𝑋)
Assertion
Ref Expression
tfr1on (𝜑𝑌 ⊆ dom 𝐹)
Distinct variable groups:   𝑓,𝐺,𝑥   𝑓,𝑋,𝑥   𝑓,𝑌,𝑥   𝜑,𝑓,𝑥
Allowed substitution hints:   𝐹(𝑥,𝑓)

Proof of Theorem tfr1on
Dummy variables 𝑎 𝑏 𝑐 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfr1on.f . 2 𝐹 = recs(𝐺)
2 tfr1on.g . 2 (𝜑 → Fun 𝐺)
3 tfr1on.x . 2 (𝜑 → Ord 𝑋)
4 tfr1on.ex . 2 ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
5 eqid 2115 . . 3 {𝑎 ∣ ∃𝑏𝑋 (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))} = {𝑎 ∣ ∃𝑏𝑋 (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))}
65tfr1onlem3 6189 . 2 {𝑎 ∣ ∃𝑏𝑋 (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))} = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
7 tfr1on.u . 2 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
8 tfr1on.yx . 2 (𝜑𝑌𝑋)
91, 2, 3, 4, 6, 7, 8tfr1onlemres 6200 1 (𝜑𝑌 ⊆ dom 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 945   = wceq 1314  wcel 1463  {cab 2101  wral 2390  wrex 2391  Vcvv 2657  wss 3037   cuni 3702  Ord word 4244  suc csuc 4247  dom cdm 4499  cres 4501  Fun wfun 5075   Fn wfn 5076  cfv 5081  recscrecs 6155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-recs 6156
This theorem is referenced by:  tfri1dALT  6202
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