Theorem tfri1d 6390

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 2193	. . . . . 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 6366	. . . . 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 6388	. . . 4 |- ( ph -> dom recs ( G ) = On )
5		eqid 2193	. . . . 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 6372	. . . 4 |- Fun recs ( G )
7	4, 6	jctil 312	. . 3 |- ( ph -> ( Fun recs ( G ) /\ dom recs ( G ) = On ) )
8		df-fn 5258	. . 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 5349	. 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 1362   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   _Vcvv 2760   Oncon0 4395   dom cdm 4660   |` cres 4662   Fun wfun 5249   Fn wfn 5250   ` cfv 5255  recscrecs 6359

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6360

This theorem is referenced by:  tfri2d  6391  tfri1  6420  rdgifnon  6434  rdgifnon2  6435  frecfnom  6456