Theorem tfri2d 6315

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 6344). 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
tfri1d.1	|- F = recs ( G )
tfri1d.2	|- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) )

Assertion

Ref	Expression
tfri2d	|- ( ( ph /\ A e. On ) -> ( F ` A ) = ( G ` ( F |` A ) ) )

Distinct variable group:   x, G

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

Proof of Theorem tfri2d

Step	Hyp	Ref	Expression
1		tfri1d.1	. . . . . 6 |- F = recs ( G )
2		tfri1d.2	. . . . . 6 |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) )
3	1, 2	tfri1d 6314	. . . . 5 |- ( ph -> F Fn On )
4		fndm 5297	. . . . 5 |- ( F Fn On -> dom F = On )
5	3, 4	syl 14	. . . 4 |- ( ph -> dom F = On )
6	5	eleq2d 2240	. . 3 |- ( ph -> ( A e. dom F <-> A e. On ) )
7	6	biimpar 295	. 2 |- ( ( ph /\ A e. On ) -> A e. dom F )
8	1	tfr2a 6300	. 2 |- ( A e. dom F -> ( F ` A ) = ( G ` ( F |` A ) ) )
9	7, 8	syl 14	1 |- ( ( ph /\ A e. On ) -> ( F ` A ) = ( G ` ( F |` A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   = wceq 1348   e. wcel 2141   _Vcvv 2730   Oncon0 4348   dom cdm 4611   |` cres 4613   Fun wfun 5192   Fn wfn 5193   ` cfv 5198  recscrecs 6283

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-recs 6284

This theorem is referenced by:  rdgivallem  6360