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Theorem tfri2 6381
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 6380). 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
tfri1.1 𝐹 = recs(𝐺)
tfri1.2 (Fun 𝐺 ∧ (𝐺𝑥) ∈ V)
Assertion
Ref Expression
tfri2 (𝐴 ∈ On → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
Distinct variable group:   𝑥,𝐺
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem tfri2
StepHypRef Expression
1 tfri1.1 . . . . 5 𝐹 = recs(𝐺)
2 tfri1.2 . . . . 5 (Fun 𝐺 ∧ (𝐺𝑥) ∈ V)
31, 2tfri1 6380 . . . 4 𝐹 Fn On
4 fndm 5327 . . . 4 (𝐹 Fn On → dom 𝐹 = On)
53, 4ax-mp 5 . . 3 dom 𝐹 = On
65eleq2i 2254 . 2 (𝐴 ∈ dom 𝐹𝐴 ∈ On)
71tfr2a 6336 . 2 (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
86, 7sylbir 135 1 (𝐴 ∈ On → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  wcel 2158  Vcvv 2749  Oncon0 4375  dom cdm 4638  cres 4640  Fun wfun 5222   Fn wfn 5223  cfv 5228  recscrecs 6319
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 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
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 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-recs 6320
This theorem is referenced by:  tfri3  6382
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