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Theorem tfr2 7256
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 7255 . . . 4 𝐹 Fn On
3 fndm 5789 . . . 4 (𝐹 Fn On → dom 𝐹 = On)
42, 3ax-mp 5 . . 3 dom 𝐹 = On
54eleq2i 2584 . 2 (𝐴 ∈ dom 𝐹𝐴 ∈ On)
61tfr2a 7253 . 2 (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
75, 6sylbir 223 1 (𝐴 ∈ On → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1938  dom cdm 4932  cres 4934  Oncon0 5530   Fn wfn 5684  cfv 5689  recscrecs 7229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-wrecs 7168  df-recs 7230
This theorem is referenced by:  tfr3  7257  recsval  7262  rdgval  7278  dfac8alem  8610  dfac12lem1  8723  zorn2lem1  9076  ttukeylem3  9091
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