MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfr2 Structured version   Visualization version   GIF version

Theorem tfr2 8229
Description: Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47. Here we show that the function 𝐹 has the property that for any function 𝐺 whatsoever, the "next" value of 𝐹 is 𝐺 recursively applied to all "previous" values of 𝐹. (Contributed by NM, 9-Apr-1995.) (Revised by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
tfr.1 𝐹 = recs(𝐺)
Assertion
Ref Expression
tfr2 (𝐴 ∈ On → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))

Proof of Theorem tfr2
StepHypRef Expression
1 tfr.1 . . . . 5 𝐹 = recs(𝐺)
21tfr1 8228 . . . 4 𝐹 Fn On
32fndmi 6537 . . 3 dom 𝐹 = On
43eleq2i 2830 . 2 (𝐴 ∈ dom 𝐹𝐴 ∈ On)
51tfr2a 8226 . 2 (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
64, 5sylbir 234 1 (𝐴 ∈ On → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  dom cdm 5589  cres 5591  Oncon0 6266  cfv 6433  recscrecs 8201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202
This theorem is referenced by:  tfr3  8230  recsval  8235  rdgval  8251  rdg0n  8265  dfac8alem  9785  dfac12lem1  9899  zorn2lem1  10252  ttukeylem3  10267  madeval  34036
  Copyright terms: Public domain W3C validator