Theorem tfri1d 6314

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 2170	. . . . . 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 6290	. . . . 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 6312	. . . 4 |- ( ph -> dom recs ( G ) = On )
5		eqid 2170	. . . . 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 6296	. . . 4 |- Fun recs ( G )
7	4, 6	jctil 310	. . 3 |- ( ph -> ( Fun recs ( G ) /\ dom recs ( G ) = On ) )
8		df-fn 5201	. . 3 |- (recs ( G ) Fn On <-> ( Fun recs ( G ) /\ dom recs ( G ) = On ) )
9	7, 8	sylibr 133	. 2 |- ( ph -> recs ( G ) Fn On )
10		tfri1d.1	. . 3 |- F = recs ( G )
11	10	fneq1i 5292	. 2 |- ( F Fn On <-> recs ( G ) Fn On )
12	9, 11	sylibr 133	1 |- ( ph -> F Fn On )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   _Vcvv 2730   Oncon0 4348   dom cdm 4611   |` cres 4613   Fun wfun 5192   Fn wfn 5193   ` cfv 5198  recscrecs 6283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-recs 6284

This theorem is referenced by:  tfri2d  6315  tfri1  6344  rdgifnon  6358  rdgifnon2  6359  frecfnom  6380