Theorem tfri2 6419

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 6418). 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 6418	. . . 4 |- F Fn On
4		fndm 5353	. . . 4 |- ( F Fn On -> dom F = On )
5	3, 4	ax-mp 5	. . 3 |- dom F = On
6	5	eleq2i 2260	. 2 |- ( A e. dom F <-> A e. On )
7	1	tfr2a 6374	. 2 |- ( A e. dom F -> ( F ` A ) = ( G ` ( F |` A ) ) )
8	6, 7	sylbir 135	1 |- ( A e. On -> ( F ` A ) = ( G ` ( F |` A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   Oncon0 4394   dom cdm 4659   |` cres 4661   Fun wfun 5248   Fn wfn 5249   ` cfv 5254  recscrecs 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-recs 6358

This theorem is referenced by:  tfri3  6420