Theorem tfri2 6313

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 6312). 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 /\ ( G ` x ) e. _V )

Assertion

Ref	Expression
tfri2	|- ( A e. On -> ( F ` A ) = ( G ` ( F |` A ) ) )

Distinct variable group:   x, G

Allowed substitution hints:   A( x)   F( x)

Proof of Theorem tfri2

Step	Hyp	Ref	Expression
1		tfri1.1	. . . . 5 |- F = recs ( G )
2		tfri1.2	. . . . 5 |- ( Fun G /\ ( G ` x ) e. _V )
3	1, 2	tfri1 6312	. . . 4 |- F Fn On
4		fndm 5269	. . . 4 |- ( F Fn On -> dom F = On )
5	3, 4	ax-mp 5	. . 3 |- dom F = On
6	5	eleq2i 2224	. 2 |- ( A e. dom F <-> A e. On )
7	1	tfr2a 6268	. 2 |- ( A e. dom F -> ( F ` A ) = ( G ` ( F |` A ) ) )
8	6, 7	sylbir 134	1 |- ( A e. On -> ( F ` A ) = ( G ` ( F |` A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   _Vcvv 2712   Oncon0 4323   dom cdm 4586   |` cres 4588   Fun wfun 5164   Fn wfn 5165   ` cfv 5170  recscrecs 6251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-recs 6252

This theorem is referenced by:  tfri3  6314