Theorem tfri2d 6597

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 6626). 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 6596	. . . . 5 |- ( ph -> F Fn On )
4		fndm 5475	. . . . 5 |- ( F Fn On -> dom F = On )
5	3, 4	syl 14	. . . 4 |- ( ph -> dom F = On )
6	5	eleq2d 2308	. . 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 6582	. 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 1400   = wceq 1402   e. wcel 2209   _Vcvv 2821   Oncon0 4503   dom cdm 4769   |` cres 4771   Fun wfun 5366   Fn wfn 5367   ` cfv 5372  recscrecs 6565

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-recs 6566

This theorem is referenced by:  rdgivallem  6642