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Theorem tfri2d 5872
 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 5873). 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 (φx(Fun 𝐺 (𝐺x) V))
Assertion
Ref Expression
tfri2d ((φ A 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 (φx(Fun 𝐺 (𝐺x) V))
31, 2tfri1d 5871 . . . . 5 (φ𝐹 Fn On)
4 fndm 4924 . . . . 5 (𝐹 Fn On → dom 𝐹 = On)
53, 4syl 14 . . . 4 (φ → dom 𝐹 = On)
65eleq2d 2089 . . 3 (φ → (A dom 𝐹A On))
76biimpar 281 . 2 ((φ A On) → A dom 𝐹)
81tfr2a 5858 . 2 (A dom 𝐹 → (𝐹A) = (𝐺‘(𝐹A)))
97, 8syl 14 1 ((φ A On) → (𝐹A) = (𝐺‘(𝐹A)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1226   = wceq 1228   ∈ wcel 1374  Vcvv 2535  Oncon0 4049  dom cdm 4272   ↾ cres 4274  Fun wfun 4823   Fn wfn 4824  ‘cfv 4829  recscrecs 5841 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-recs 5842 This theorem is referenced by:  rdgi0g  5886  rdgivallem  5888  frec0g  5902  frecsuclem1  5903
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