Theorem tfr1on 6611

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 2238	. . 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 6599	. 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 6610	1 |- ( ph -> Y C_ dom F )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   _Vcvv 2821   C_ wss 3220   U.cuni 3930   Ord word 4502   suc csuc 4505   dom cdm 4769   |` cres 4771   Fun wfun 5366   Fn wfn 5367   ` cfv 5372  recscrecs 6565

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-recs 6566

This theorem is referenced by:  tfri1dALT  6612