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Theorem tfr1on 6063
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 2085 . . 3 {𝑎 ∣ ∃𝑏𝑋 (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))} = {𝑎 ∣ ∃𝑏𝑋 (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))}
65tfr1onlem3 6051 . 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 6062 1 (𝜑𝑌 ⊆ dom 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 922   = wceq 1287  wcel 1436  {cab 2071  wral 2355  wrex 2356  Vcvv 2615  wss 2988   cuni 3636  Ord word 4162  suc csuc 4165  dom cdm 4410  cres 4412  Fun wfun 4972   Fn wfn 4973  cfv 4978  recscrecs 6017
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3928  ax-sep 3931  ax-pow 3983  ax-pr 4009  ax-un 4233  ax-setind 4325
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3911  df-id 4093  df-iord 4166  df-on 4168  df-suc 4171  df-xp 4416  df-rel 4417  df-cnv 4418  df-co 4419  df-dm 4420  df-rn 4421  df-res 4422  df-ima 4423  df-iota 4943  df-fun 4980  df-fn 4981  df-f 4982  df-f1 4983  df-fo 4984  df-f1o 4985  df-fv 4986  df-recs 6018
This theorem is referenced by:  tfri1dALT  6064
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