Theorem tfr1on 6318

Description: Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 12-Mar-2022.)

Hypotheses

Ref	Expression
tfr1on.f	|- F = recs ( G )
tfr1on.g	|- ( ph -> Fun G )
tfr1on.x	|- ( ph -> Ord X )
tfr1on.ex	|- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
tfr1on.u	|- ( ( ph /\ x e. U. X ) -> suc x e. X )
tfr1on.yx	|- ( ph -> Y e. X )

Assertion

Ref	Expression
tfr1on	|- ( ph -> Y C_ dom F )

Distinct variable groups:   f, G, x   f, X, x   f, Y, x   ph, f, x

Allowed substitution hints:   F( x, f)

Proof of Theorem tfr1on

Dummy variables a b c y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfr1on.f	. 2 |- F = recs ( G )
2		tfr1on.g	. 2 |- ( ph -> Fun G )
3		tfr1on.x	. 2 |- ( ph -> Ord X )
4		tfr1on.ex	. 2 |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
5		eqid 2165	. . 3 |- { a | E. b e. X ( a Fn b /\ A. c e. b ( a ` c ) = ( G ` ( a |` c ) ) ) } = { a | E. b e. X ( a Fn b /\ A. c e. b ( a ` c ) = ( G ` ( a |` c ) ) ) }
6	5	tfr1onlem3 6306	. 2 |- { a | E. b e. X ( a Fn b /\ A. c e. b ( a ` c ) = ( G ` ( a |` c ) ) ) } = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
7		tfr1on.u	. 2 |- ( ( ph /\ x e. U. X ) -> suc x e. X )
8		tfr1on.yx	. 2 |- ( ph -> Y e. X )
9	1, 2, 3, 4, 6, 7, 8	tfr1onlemres 6317	1 |- ( ph -> Y C_ dom F )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   _Vcvv 2726   C_ wss 3116   U.cuni 3789   Ord word 4340   suc csuc 4343   dom cdm 4604   |` cres 4606   Fun wfun 5182   Fn wfn 5183   ` cfv 5188  recscrecs 6272

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-recs 6273

This theorem is referenced by:  tfri1dALT  6319