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Theorem tfri2d 5950
 Description: Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule 𝐺 ( as described at tfri1 5951). Here we show that the function 𝐹 has the property that for any function 𝐺 satisfying that condition, the "next" value of 𝐹 is 𝐺 recursively applied to all "previous" values of 𝐹. (Contributed by Jim Kingdon, 4-May-2019.)
Hypotheses
Ref Expression
tfri1d.1 𝐹 = recs(𝐺)
tfri1d.2 (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))
Assertion
Ref Expression
tfri2d ((𝜑𝐴 ∈ On) → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
Distinct variable group:   𝑥,𝐺
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem tfri2d
StepHypRef Expression
1 tfri1d.1 . . . . . 6 𝐹 = recs(𝐺)
2 tfri1d.2 . . . . . 6 (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))
31, 2tfri1d 5949 . . . . 5 (𝜑𝐹 Fn On)
4 fndm 4998 . . . . 5 (𝐹 Fn On → dom 𝐹 = On)
53, 4syl 14 . . . 4 (𝜑 → dom 𝐹 = On)
65eleq2d 2107 . . 3 (𝜑 → (𝐴 ∈ dom 𝐹𝐴 ∈ On))
76biimpar 281 . 2 ((𝜑𝐴 ∈ On) → 𝐴 ∈ dom 𝐹)
81tfr2a 5936 . 2 (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
97, 8syl 14 1 ((𝜑𝐴 ∈ On) → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557  Oncon0 4100  dom cdm 4345   ↾ cres 4347  Fun wfun 4896   Fn wfn 4897  ‘cfv 4902  recscrecs 5919 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920 This theorem is referenced by:  rdgivallem  5968  frecsuclem1  5987
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