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Theorem tfr2 8008
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 8007 . . . 4 𝐹 Fn On
3 fndm 6427 . . . 4 (𝐹 Fn On → dom 𝐹 = On)
42, 3ax-mp 5 . . 3 dom 𝐹 = On
54eleq2i 2902 . 2 (𝐴 ∈ dom 𝐹𝐴 ∈ On)
61tfr2a 8005 . 2 (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
75, 6sylbir 237 1 (𝐴 ∈ On → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  dom cdm 5527  cres 5529  Oncon0 6163   Fn wfn 6322  cfv 6327  recscrecs 7981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5162  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4811  df-iun 4893  df-br 5039  df-opab 5101  df-mpt 5119  df-tr 5145  df-id 5432  df-eprel 5437  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6120  df-ord 6166  df-on 6167  df-suc 6169  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-wrecs 7921  df-recs 7982
This theorem is referenced by:  tfr3  8009  recsval  8014  rdgval  8030  dfac8alem  9429  dfac12lem1  9543  zorn2lem1  9892  ttukeylem3  9907  madeval  33293
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