Theorem tfri1d 6479

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 2229	. . . . . 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 6455	. . . . 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 6477	. . . 4 |- ( ph -> dom recs ( G ) = On )
5		eqid 2229	. . . . 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 6461	. . . 4 |- Fun recs ( G )
7	4, 6	jctil 312	. . 3 |- ( ph -> ( Fun recs ( G ) /\ dom recs ( G ) = On ) )
8		df-fn 5320	. . 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 5414	. 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 1393   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   _Vcvv 2799   Oncon0 4453   dom cdm 4718   |` cres 4720   Fun wfun 5311   Fn wfn 5312   ` cfv 5317  recscrecs 6448

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-recs 6449

This theorem is referenced by:  tfri2d  6480  tfri1  6509  rdgifnon  6523  rdgifnon2  6524  frecfnom  6545