Theorem frecrdg 5992

Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 5978 produces the same results as df-irdg 5957 restricted to om.

Presumably the theorem would also hold if F Fn _V were changed to A. z ( F ` z ) e. _V. (Contributed by Jim Kingdon, 29-Aug-2019.)

Hypotheses

Ref	Expression
frecrdg.1	|- ( ph -> F Fn _V )
frecrdg.2	|- ( ph -> A e. V )
frecrdg.inc	|- ( ph -> A. x x C_ ( F ` x ) )

Assertion

Ref	Expression
frecrdg	|- ( ph -> frec ( F , A ) = ( rec ( F , A ) |` om ) )

Distinct variable groups:   x, A   x, F   x, V   ph, x

Proof of Theorem frecrdg

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frecrdg.1	. . . 4 |- ( ph -> F Fn _V )
2		vex 2560	. . . . . 6 |- z e. _V
3		funfvex 5192	. . . . . . 7 |- ( ( Fun F /\ z e. dom F ) -> ( F ` z ) e. _V )
4	3	funfni 4999	. . . . . 6 |- ( ( F Fn _V /\ z e. _V ) -> ( F ` z ) e. _V )
5	2, 4	mpan2 401	. . . . 5 |- ( F Fn _V -> ( F ` z ) e. _V )
6	5	alrimiv 1754	. . . 4 |- ( F Fn _V -> A. z ( F ` z ) e. _V )
7	1, 6	syl 14	. . 3 |- ( ph -> A. z ( F ` z ) e. _V )
8		frecrdg.2	. . 3 |- ( ph -> A e. V )
9		frecfnom 5986	. . 3 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> frec ( F , A ) Fn om )
10	7, 8, 9	syl2anc 391	. 2 |- ( ph -> frec ( F , A ) Fn om )
11		rdgifnon2 5967	. . . 4 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> rec ( F , A ) Fn On )
12	7, 8, 11	syl2anc 391	. . 3 |- ( ph -> rec ( F , A ) Fn On )
13		omsson 4335	. . 3 |- om C_ On
14		fnssres 5012	. . 3 |- ( ( rec ( F , A ) Fn On /\ om C_ On ) -> ( rec ( F , A ) |` om ) Fn om )
15	12, 13, 14	sylancl 392	. 2 |- ( ph -> ( rec ( F , A ) |` om ) Fn om )
16		fveq2 5178	. . . . 5 |- ( x = (/) -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` (/) ) )
17		fveq2 5178	. . . . 5 |- ( x = (/) -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` (/) ) )
18	16, 17	eqeq12d 2054	. . . 4 |- ( x = (/) -> ( (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) <-> (frec ( F , A ) ` (/) ) = ( ( rec ( F , A ) |` om ) ` (/) ) ) )
19		fveq2 5178	. . . . 5 |- ( x = y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` y ) )
20		fveq2 5178	. . . . 5 |- ( x = y -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` y ) )
21	19, 20	eqeq12d 2054	. . . 4 |- ( x = y -> ( (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) <-> (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) )
22		fveq2 5178	. . . . 5 |- ( x = suc y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` suc y ) )
23		fveq2 5178	. . . . 5 |- ( x = suc y -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` suc y ) )
24	22, 23	eqeq12d 2054	. . . 4 |- ( x = suc y -> ( (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) <-> (frec ( F , A ) ` suc y ) = ( ( rec ( F , A ) |` om ) ` suc y ) ) )
25		frec0g 5983	. . . . . 6 |- ( A e. V -> (frec ( F , A ) ` (/) ) = A )
26	8, 25	syl 14	. . . . 5 |- ( ph -> (frec ( F , A ) ` (/) ) = A )
27		peano1 4317	. . . . . . 7 |- (/) e. om
28		fvres 5198	. . . . . . 7 |- ( (/) e. om -> ( ( rec ( F , A ) |` om ) ` (/) ) = ( rec ( F , A ) ` (/) ) )
29	27, 28	ax-mp 7	. . . . . 6 |- ( ( rec ( F , A ) |` om ) ` (/) ) = ( rec ( F , A ) ` (/) )
30		rdg0g 5975	. . . . . . 7 |- ( A e. V -> ( rec ( F , A ) ` (/) ) = A )
31	8, 30	syl 14	. . . . . 6 |- ( ph -> ( rec ( F , A ) ` (/) ) = A )
32	29, 31	syl5eq 2084	. . . . 5 |- ( ph -> ( ( rec ( F , A ) |` om ) ` (/) ) = A )
33	26, 32	eqtr4d 2075	. . . 4 |- ( ph -> (frec ( F , A ) ` (/) ) = ( ( rec ( F , A ) |` om ) ` (/) ) )
34		simpr 103	. . . . . . . . . 10 |- ( ( ( ph /\ y e. om ) /\ (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) -> (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) )
35		fvres 5198	. . . . . . . . . . 11 |- ( y e. om -> ( ( rec ( F , A ) |` om ) ` y ) = ( rec ( F , A ) ` y ) )
36	35	ad2antlr 458	. . . . . . . . . 10 |- ( ( ( ph /\ y e. om ) /\ (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) -> ( ( rec ( F , A ) |` om ) ` y ) = ( rec ( F , A ) ` y ) )
37	34, 36	eqtrd 2072	. . . . . . . . 9 |- ( ( ( ph /\ y e. om ) /\ (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) -> (frec ( F , A ) ` y ) = ( rec ( F , A ) ` y ) )
38	37	fveq2d 5182	. . . . . . . 8 |- ( ( ( ph /\ y e. om ) /\ (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) -> ( F ` (frec ( F , A ) ` y ) ) = ( F ` ( rec ( F , A ) ` y ) ) )
39	7, 8	jca 290	. . . . . . . . . 10 |- ( ph -> ( A. z ( F ` z ) e. _V /\ A e. V ) )
40		frecsuc 5991	. . . . . . . . . . 11 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> (frec ( F , A ) ` suc y ) = ( F ` (frec ( F , A ) ` y ) ) )
41	40	3expa 1104	. . . . . . . . . 10 |- ( ( ( A. z ( F ` z ) e. _V /\ A e. V ) /\ y e. om ) -> (frec ( F , A ) ` suc y ) = ( F ` (frec ( F , A ) ` y ) ) )
42	39, 41	sylan 267	. . . . . . . . 9 |- ( ( ph /\ y e. om ) -> (frec ( F , A ) ` suc y ) = ( F ` (frec ( F , A ) ` y ) ) )
43	42	adantr 261	. . . . . . . 8 |- ( ( ( ph /\ y e. om ) /\ (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) -> (frec ( F , A ) ` suc y ) = ( F ` (frec ( F , A ) ` y ) ) )
44	1	adantr 261	. . . . . . . . . 10 |- ( ( ph /\ y e. om ) -> F Fn _V )
45	8	adantr 261	. . . . . . . . . 10 |- ( ( ph /\ y e. om ) -> A e. V )
46		simpr 103	. . . . . . . . . . 11 |- ( ( ph /\ y e. om ) -> y e. om )
47		nnon 4332	. . . . . . . . . . 11 |- ( y e. om -> y e. On )
48	46, 47	syl 14	. . . . . . . . . 10 |- ( ( ph /\ y e. om ) -> y e. On )
49		frecrdg.inc	. . . . . . . . . . 11 |- ( ph -> A. x x C_ ( F ` x ) )
50	49	adantr 261	. . . . . . . . . 10 |- ( ( ph /\ y e. om ) -> A. x x C_ ( F ` x ) )
51	44, 45, 48, 50	rdgisucinc 5972	. . . . . . . . 9 |- ( ( ph /\ y e. om ) -> ( rec ( F , A ) ` suc y ) = ( F ` ( rec ( F , A ) ` y ) ) )
52	51	adantr 261	. . . . . . . 8 |- ( ( ( ph /\ y e. om ) /\ (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) -> ( rec ( F , A ) ` suc y ) = ( F ` ( rec ( F , A ) ` y ) ) )
53	38, 43, 52	3eqtr4d 2082	. . . . . . 7 |- ( ( ( ph /\ y e. om ) /\ (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) -> (frec ( F , A ) ` suc y ) = ( rec ( F , A ) ` suc y ) )
54		peano2 4318	. . . . . . . . 9 |- ( y e. om -> suc y e. om )
55		fvres 5198	. . . . . . . . 9 |- ( suc y e. om -> ( ( rec ( F , A ) |` om ) ` suc y ) = ( rec ( F , A ) ` suc y ) )
56	54, 55	syl 14	. . . . . . . 8 |- ( y e. om -> ( ( rec ( F , A ) |` om ) ` suc y ) = ( rec ( F , A ) ` suc y ) )
57	56	ad2antlr 458	. . . . . . 7 |- ( ( ( ph /\ y e. om ) /\ (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) -> ( ( rec ( F , A ) |` om ) ` suc y ) = ( rec ( F , A ) ` suc y ) )
58	53, 57	eqtr4d 2075	. . . . . 6 |- ( ( ( ph /\ y e. om ) /\ (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) -> (frec ( F , A ) ` suc y ) = ( ( rec ( F , A ) |` om ) ` suc y ) )
59	58	ex 108	. . . . 5 |- ( ( ph /\ y e. om ) -> ( (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) -> (frec ( F , A ) ` suc y ) = ( ( rec ( F , A ) |` om ) ` suc y ) ) )
60	59	expcom 109	. . . 4 |- ( y e. om -> ( ph -> ( (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) -> (frec ( F , A ) ` suc y ) = ( ( rec ( F , A ) |` om ) ` suc y ) ) ) )
61	18, 21, 24, 33, 60	finds2 4324	. . 3 |- ( x e. om -> ( ph -> (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) ) )
62	61	impcom 116	. 2 |- ( ( ph /\ x e. om ) -> (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) )
63	10, 15, 62	eqfnfvd 5268	1 |- ( ph -> frec ( F , A ) = ( rec ( F , A ) |` om ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   e. wcel 1393   _Vcvv 2557   C_ wss 2917   (/)c0 3224   Oncon0 4100   suc csuc 4102   omcom 4313   |` cres 4347   Fn wfn 4897   ` cfv 4902   reccrdg 5956  freccfrec 5977

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-irdg 5957  df-frec 5978

This theorem is referenced by: (None)