Theorem tfri1d 5949

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" and here is stated as ( G ` x ) e. _V. Alternatively 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 2040	. . . . . 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 5926	. . . . 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 5947	. . . 4 |- ( ph -> dom recs ( G ) = On )
5		eqid 2040	. . . . 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 5933	. . . 4 |- Fun recs ( G )
7	4, 6	jctil 295	. . 3 |- ( ph -> ( Fun recs ( G ) /\ dom recs ( G ) = On ) )
8		df-fn 4905	. . 3 |- (recs ( G ) Fn On <-> ( Fun recs ( G ) /\ dom recs ( G ) = On ) )
9	7, 8	sylibr 137	. 2 |- ( ph -> recs ( G ) Fn On )
10		tfri1d.1	. . 3 |- F = recs ( G )
11	10	fneq1i 4993	. 2 |- ( F Fn On <-> recs ( G ) Fn On )
12	9, 11	sylibr 137	1 |- ( ph -> F Fn On )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   _Vcvv 2557   Oncon0 4100   dom cdm 4345   |` cres 4347   Fun wfun 4896   Fn wfn 4897   ` cfv 4902  recscrecs 5919

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920

This theorem is referenced by:  tfri2d  5950  tfri1  5951  rdgifnon  5966  rdgifnon2  5967  frecfnom  5986  frecsuclemdm  5988  frecsuclem3  5990