Theorem frecrdg 6298

Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6281 produces the same results as df-irdg 6260 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 2684	. . . . . 6 |- z e. _V
3		funfvex 5431	. . . . . . 7 |- ( ( Fun F /\ z e. dom F ) -> ( F ` z ) e. _V )
4	3	funfni 5218	. . . . . 6 |- ( ( F Fn _V /\ z e. _V ) -> ( F ` z ) e. _V )
5	2, 4	mpan2 421	. . . . 5 |- ( F Fn _V -> ( F ` z ) e. _V )
6	5	alrimiv 1846	. . . 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 6291	. . 3 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> frec ( F , A ) Fn om )
10	7, 8, 9	syl2anc 408	. 2 |- ( ph -> frec ( F , A ) Fn om )
11		rdgifnon2 6270	. . . 4 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> rec ( F , A ) Fn On )
12	7, 8, 11	syl2anc 408	. . 3 |- ( ph -> rec ( F , A ) Fn On )
13		omsson 4521	. . 3 |- om C_ On
14		fnssres 5231	. . 3 |- ( ( rec ( F , A ) Fn On /\ om C_ On ) -> ( rec ( F , A ) |` om ) Fn om )
15	12, 13, 14	sylancl 409	. 2 |- ( ph -> ( rec ( F , A ) |` om ) Fn om )
16		fveq2 5414	. . . . 5 |- ( x = (/) -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` (/) ) )
17		fveq2 5414	. . . . 5 |- ( x = (/) -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` (/) ) )
18	16, 17	eqeq12d 2152	. . . 4 |- ( x = (/) -> ( (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) <-> (frec ( F , A ) ` (/) ) = ( ( rec ( F , A ) |` om ) ` (/) ) ) )
19		fveq2 5414	. . . . 5 |- ( x = y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` y ) )
20		fveq2 5414	. . . . 5 |- ( x = y -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` y ) )
21	19, 20	eqeq12d 2152	. . . 4 |- ( x = y -> ( (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) <-> (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) )
22		fveq2 5414	. . . . 5 |- ( x = suc y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` suc y ) )
23		fveq2 5414	. . . . 5 |- ( x = suc y -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` suc y ) )
24	22, 23	eqeq12d 2152	. . . 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 6287	. . . . . 6 |- ( A e. V -> (frec ( F , A ) ` (/) ) = A )
26	8, 25	syl 14	. . . . 5 |- ( ph -> (frec ( F , A ) ` (/) ) = A )
27		peano1 4503	. . . . . . 7 |- (/) e. om
28		fvres 5438	. . . . . . 7 |- ( (/) e. om -> ( ( rec ( F , A ) |` om ) ` (/) ) = ( rec ( F , A ) ` (/) ) )
29	27, 28	ax-mp 5	. . . . . 6 |- ( ( rec ( F , A ) |` om ) ` (/) ) = ( rec ( F , A ) ` (/) )
30		rdg0g 6278	. . . . . . 7 |- ( A e. V -> ( rec ( F , A ) ` (/) ) = A )
31	8, 30	syl 14	. . . . . 6 |- ( ph -> ( rec ( F , A ) ` (/) ) = A )
32	29, 31	syl5eq 2182	. . . . 5 |- ( ph -> ( ( rec ( F , A ) |` om ) ` (/) ) = A )
33	26, 32	eqtr4d 2173	. . . 4 |- ( ph -> (frec ( F , A ) ` (/) ) = ( ( rec ( F , A ) |` om ) ` (/) ) )
34		simpr 109	. . . . . . . . . 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 5438	. . . . . . . . . . 11 |- ( y e. om -> ( ( rec ( F , A ) |` om ) ` y ) = ( rec ( F , A ) ` y ) )
36	35	ad2antlr 480	. . . . . . . . . 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 2170	. . . . . . . . 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 5418	. . . . . . . 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 304	. . . . . . . . . 10 |- ( ph -> ( A. z ( F ` z ) e. _V /\ A e. V ) )
40		simp1 981	. . . . . . . . . . . . 13 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A. z ( F ` z ) e. _V )
41		ralv 2698	. . . . . . . . . . . . 13 |- ( A. z e. _V ( F ` z ) e. _V <-> A. z ( F ` z ) e. _V )
42	40, 41	sylibr 133	. . . . . . . . . . . 12 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A. z e. _V ( F ` z ) e. _V )
43		simp2 982	. . . . . . . . . . . . 13 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A e. V )
44	43	elexd 2694	. . . . . . . . . . . 12 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A e. _V )
45		simp3 983	. . . . . . . . . . . 12 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> y e. om )
46		frecsuc 6297	. . . . . . . . . . . 12 |- ( ( A. z e. _V ( F ` z ) e. _V /\ A e. _V /\ y e. om ) -> (frec ( F , A ) ` suc y ) = ( F ` (frec ( F , A ) ` y ) ) )
47	42, 44, 45, 46	syl3anc 1216	. . . . . . . . . . 11 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> (frec ( F , A ) ` suc y ) = ( F ` (frec ( F , A ) ` y ) ) )
48	47	3expa 1181	. . . . . . . . . 10 |- ( ( ( A. z ( F ` z ) e. _V /\ A e. V ) /\ y e. om ) -> (frec ( F , A ) ` suc y ) = ( F ` (frec ( F , A ) ` y ) ) )
49	39, 48	sylan 281	. . . . . . . . 9 |- ( ( ph /\ y e. om ) -> (frec ( F , A ) ` suc y ) = ( F ` (frec ( F , A ) ` y ) ) )
50	49	adantr 274	. . . . . . . 8 |- ( ( ( ph /\ y e. om ) /\ (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) -> (frec ( F , A ) ` suc y ) = ( F ` (frec ( F , A ) ` y ) ) )
51	1	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ y e. om ) -> F Fn _V )
52	8	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ y e. om ) -> A e. V )
53		simpr 109	. . . . . . . . . . 11 |- ( ( ph /\ y e. om ) -> y e. om )
54		nnon 4518	. . . . . . . . . . 11 |- ( y e. om -> y e. On )
55	53, 54	syl 14	. . . . . . . . . 10 |- ( ( ph /\ y e. om ) -> y e. On )
56		frecrdg.inc	. . . . . . . . . . 11 |- ( ph -> A. x x C_ ( F ` x ) )
57	56	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ y e. om ) -> A. x x C_ ( F ` x ) )
58	51, 52, 55, 57	rdgisucinc 6275	. . . . . . . . 9 |- ( ( ph /\ y e. om ) -> ( rec ( F , A ) ` suc y ) = ( F ` ( rec ( F , A ) ` y ) ) )
59	58	adantr 274	. . . . . . . 8 |- ( ( ( ph /\ y e. om ) /\ (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) -> ( rec ( F , A ) ` suc y ) = ( F ` ( rec ( F , A ) ` y ) ) )
60	38, 50, 59	3eqtr4d 2180	. . . . . . 7 |- ( ( ( ph /\ y e. om ) /\ (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) -> (frec ( F , A ) ` suc y ) = ( rec ( F , A ) ` suc y ) )
61		peano2 4504	. . . . . . . . 9 |- ( y e. om -> suc y e. om )
62		fvres 5438	. . . . . . . . 9 |- ( suc y e. om -> ( ( rec ( F , A ) |` om ) ` suc y ) = ( rec ( F , A ) ` suc y ) )
63	61, 62	syl 14	. . . . . . . 8 |- ( y e. om -> ( ( rec ( F , A ) |` om ) ` suc y ) = ( rec ( F , A ) ` suc y ) )
64	63	ad2antlr 480	. . . . . . 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 ) )
65	60, 64	eqtr4d 2173	. . . . . 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 ) )
66	65	ex 114	. . . . 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 ) ) )
67	66	expcom 115	. . . 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 ) ) ) )
68	18, 21, 24, 33, 67	finds2 4510	. . 3 |- ( x e. om -> ( ph -> (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) ) )
69	68	impcom 124	. 2 |- ( ( ph /\ x e. om ) -> (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) )
70	10, 15, 69	eqfnfvd 5514	1 |- ( ph -> frec ( F , A ) = ( rec ( F , A ) |` om ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   A.wal 1329   = wceq 1331   e. wcel 1480   A.wral 2414   _Vcvv 2681   C_ wss 3066   (/)c0 3358   Oncon0 4280   suc csuc 4282   omcom 4499   |` cres 4536   Fn wfn 5113   ` cfv 5118   reccrdg 6259  freccfrec 6280

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-recs 6195  df-irdg 6260  df-frec 6281

This theorem is referenced by: (None)