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Theorem tfri2 5839
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 5838). 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 5838 . . . 4 𝐹 Fn On
4 fndm 4891 . . . 4 (𝐹 Fn On → dom 𝐹 = On)
53, 4ax-mp 7 . . 3 dom 𝐹 = On
65eleq2i 2086 . 2 (A dom 𝐹A On)
71tfr2a 5823 . 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 1373   wcel 1375  Vcvv 2533  Oncon0 4024  dom cdm 4238  cres 4240  Fun wfun 4790   Fn wfn 4791  cfv 4796  recscrecs 5806
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-coll 3824  ax-sep 3827  ax-pow 3879  ax-pr 3896  ax-un 4093  ax-setind 4174
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-fal 1232  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  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 2829  df-dif 2898  df-un 2900  df-in 2902  df-ss 2909  df-nul 3203  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-iun 3611  df-br 3717  df-opab 3771  df-mpt 3772  df-tr 3807  df-id 3983  df-iord 4027  df-on 4028  df-suc 4031  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-res 4250  df-ima 4251  df-iota 4761  df-fun 4798  df-fn 4799  df-f 4800  df-f1 4801  df-fo 4802  df-f1o 4803  df-fv 4804  df-recs 5807
This theorem is referenced by:  tfri3  5840  rdg0  5861
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