ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfri2 Structured version   GIF version

Theorem tfri2 5862
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 5861). 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 5861 . . . 4 𝐹 Fn On
4 fndm 4912 . . . 4 (𝐹 Fn On → dom 𝐹 = On)
53, 4ax-mp 7 . . 3 dom 𝐹 = On
65eleq2i 2077 . 2 (A dom 𝐹A On)
71tfr2a 5846 . 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 1223   wcel 1366  Vcvv 2526  Oncon0 4038  dom cdm 4260  cres 4262  Fun wfun 4811   Fn wfn 4812  cfv 4817  recscrecs 5829
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 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-13 1377  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-coll 3835  ax-sep 3838  ax-pow 3890  ax-pr 3907  ax-un 4108  ax-setind 4192
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-fal 1229  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ne 2179  df-ral 2280  df-rex 2281  df-reu 2282  df-rab 2284  df-v 2528  df-sbc 2733  df-csb 2821  df-dif 2888  df-un 2890  df-in 2892  df-ss 2899  df-nul 3193  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-iun 3622  df-br 3728  df-opab 3782  df-mpt 3783  df-tr 3818  df-id 3993  df-iord 4041  df-on 4043  df-suc 4046  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-f 4821  df-f1 4822  df-fo 4823  df-f1o 4824  df-fv 4825  df-recs 5830
This theorem is referenced by:  tfri3  5863  rdg0  5883
  Copyright terms: Public domain W3C validator