Theorem tfr1on 6255

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 2140	. . 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 6243	. 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 6254	1 |- ( ph -> Y C_ dom F )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1332   e. wcel 1481   {cab 2126   A.wral 2417   E.wrex 2418   _Vcvv 2689   C_ wss 3076   U.cuni 3744   Ord word 4292   suc csuc 4295   dom cdm 4547   |` cres 4549   Fun wfun 5125   Fn wfn 5126   ` cfv 5131  recscrecs 6209

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-recs 6210

This theorem is referenced by:  tfri1dALT  6256