Theorem tfr1on 6459

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 2207	. . 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 6447	. 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 6458	1 |- ( ph -> Y C_ dom F )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2178   {cab 2193   A.wral 2486   E.wrex 2487   _Vcvv 2776   C_ wss 3174   U.cuni 3864   Ord word 4427   suc csuc 4430   dom cdm 4693   |` cres 4695   Fun wfun 5284   Fn wfn 5285   ` cfv 5290  recscrecs 6413

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-recs 6414

This theorem is referenced by:  tfri1dALT  6460