Theorem tfri2d 6421

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 6450). 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 6420	. . . . 5 |- ( ph -> F Fn On )
4		fndm 5372	. . . . 5 |- ( F Fn On -> dom F = On )
5	3, 4	syl 14	. . . 4 |- ( ph -> dom F = On )
6	5	eleq2d 2274	. . 3 |- ( ph -> ( A e. dom F <-> A e. On ) )
7	6	biimpar 297	. 2 |- ( ( ph /\ A e. On ) -> A e. dom F )
8	1	tfr2a 6406	. 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 104   A.wal 1370   = wceq 1372   e. wcel 2175   _Vcvv 2771   Oncon0 4409   dom cdm 4674   |` cres 4676   Fun wfun 5264   Fn wfn 5265   ` cfv 5270  recscrecs 6389

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-recs 6390

This theorem is referenced by:  rdgivallem  6466