Theorem tfri2 6531

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 6530). 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 6530	. . . 4 |- F Fn On
4		fndm 5429	. . . 4 |- ( F Fn On -> dom F = On )
5	3, 4	ax-mp 5	. . 3 |- dom F = On
6	5	eleq2i 2298	. 2 |- ( A e. dom F <-> A e. On )
7	1	tfr2a 6486	. 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 1397   e. wcel 2202   _Vcvv 2802   Oncon0 4460   dom cdm 4725   |` cres 4727   Fun wfun 5320   Fn wfn 5321   ` cfv 5326  recscrecs 6469

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-recs 6470

This theorem is referenced by:  tfri3  6532