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Theorem tfri2 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 5871). 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 5871 . . . 4 𝐹 Fn On
4 fndm 4922 . . . 4 (𝐹 Fn On → dom 𝐹 = On)
53, 4ax-mp 7 . . 3 dom 𝐹 = On
65eleq2i 2086 . 2 (A dom 𝐹A On)
71tfr2a 5856 . 2 (A dom 𝐹 → (𝐹A) = (𝐺‘(𝐹A)))
86, 7sylbir 125 1 (A On → (𝐹A) = (𝐺‘(𝐹A)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228   wcel 1374  Vcvv 2533  Oncon0 4047  dom cdm 4270  cres 4272  Fun wfun 4821   Fn wfn 4822  cfv 4827  recscrecs 5839
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 3844  ax-sep 3847  ax-pow 3899  ax-pr 3916  ax-un 4118  ax-setind 4202
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 2287  df-rex 2288  df-reu 2289  df-rab 2291  df-v 2535  df-sbc 2740  df-csb 2828  df-dif 2895  df-un 2897  df-in 2899  df-ss 2906  df-nul 3200  df-pw 3334  df-sn 3354  df-pr 3355  df-op 3357  df-uni 3553  df-iun 3631  df-br 3737  df-opab 3791  df-mpt 3792  df-tr 3827  df-id 4002  df-iord 4050  df-on 4052  df-suc 4055  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-res 4282  df-ima 4283  df-iota 4792  df-fun 4829  df-fn 4830  df-f 4831  df-f1 4832  df-fo 4833  df-f1o 4834  df-fv 4835  df-recs 5840
This theorem is referenced by:  tfri3  5873  rdg0  5893
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