Theorem tfri2d 6440

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 6469). 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 6439	. . . . 5 |- ( ph -> F Fn On )
4		fndm 5387	. . . . 5 |- ( F Fn On -> dom F = On )
5	3, 4	syl 14	. . . 4 |- ( ph -> dom F = On )
6	5	eleq2d 2276	. . 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 6425	. 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 1371   = wceq 1373   e. wcel 2177   _Vcvv 2773   Oncon0 4423   dom cdm 4688   |` cres 4690   Fun wfun 5279   Fn wfn 5280   ` cfv 5285  recscrecs 6408

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-recs 6409

This theorem is referenced by:  rdgivallem  6485