Theorem tfri1d 6500

Description: Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition.

The condition is that G is defined "everywhere", which is stated here as ( G ` x ) e. _V. Alternately, A. x e. On A. f ( f Fn x -> f e. dom G ) would suffice.

Given a function G satisfying that condition, we define a class A of all "acceptable" functions. The final function we're interested in is the union F = recs ( G ) of them. F is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of F. In this first part we show that F is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)

Hypotheses

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

Assertion

Ref	Expression
tfri1d	|- ( ph -> F Fn On )

Distinct variable group:   x, G

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

Proof of Theorem tfri1d

Dummy variables f g u w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2231	. . . . . 6 |- { g | E. z e. On ( g Fn z /\ A. u e. z ( g ` u ) = ( G ` ( g |` u ) ) ) } = { g | E. z e. On ( g Fn z /\ A. u e. z ( g ` u ) = ( G ` ( g |` u ) ) ) }
2	1	tfrlem3 6476	. . . . 5 |- { g | E. z e. On ( g Fn z /\ A. u e. z ( g ` u ) = ( G ` ( g |` u ) ) ) } = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
3		tfri1d.2	. . . . 5 |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) )
4	2, 3	tfrlemi14d 6498	. . . 4 |- ( ph -> dom recs ( G ) = On )
5		eqid 2231	. . . . 5 |- { w | E. y e. On ( w Fn y /\ A. z e. y ( w ` z ) = ( G ` ( w |` z ) ) ) } = { w | E. y e. On ( w Fn y /\ A. z e. y ( w ` z ) = ( G ` ( w |` z ) ) ) }
6	5	tfrlem7 6482	. . . 4 |- Fun recs ( G )
7	4, 6	jctil 312	. . 3 |- ( ph -> ( Fun recs ( G ) /\ dom recs ( G ) = On ) )
8		df-fn 5329	. . 3 |- (recs ( G ) Fn On <-> ( Fun recs ( G ) /\ dom recs ( G ) = On ) )
9	7, 8	sylibr 134	. 2 |- ( ph -> recs ( G ) Fn On )
10		tfri1d.1	. . 3 |- F = recs ( G )
11	10	fneq1i 5424	. 2 |- ( F Fn On <-> recs ( G ) Fn On )
12	9, 11	sylibr 134	1 |- ( ph -> F Fn On )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1395   = wceq 1397   e. wcel 2202   {cab 2217   A.wral 2510   E.wrex 2511   _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:  tfri2d  6501  tfri1  6530  rdgifnon  6544  rdgifnon2  6545  frecfnom  6566