Theorem frecrdg 6554

Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6537 produces the same results as df-irdg 6516 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 2802	. . . . . 6 |- z e. _V
3		funfvex 5644	. . . . . . 7 |- ( ( Fun F /\ z e. dom F ) -> ( F ` z ) e. _V )
4	3	funfni 5423	. . . . . 6 |- ( ( F Fn _V /\ z e. _V ) -> ( F ` z ) e. _V )
5	2, 4	mpan2 425	. . . . 5 |- ( F Fn _V -> ( F ` z ) e. _V )
6	5	alrimiv 1920	. . . 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 6547	. . 3 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> frec ( F , A ) Fn om )
10	7, 8, 9	syl2anc 411	. 2 |- ( ph -> frec ( F , A ) Fn om )
11		rdgifnon2 6526	. . . 4 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> rec ( F , A ) Fn On )
12	7, 8, 11	syl2anc 411	. . 3 |- ( ph -> rec ( F , A ) Fn On )
13		omsson 4705	. . 3 |- om C_ On
14		fnssres 5436	. . 3 |- ( ( rec ( F , A ) Fn On /\ om C_ On ) -> ( rec ( F , A ) |` om ) Fn om )
15	12, 13, 14	sylancl 413	. 2 |- ( ph -> ( rec ( F , A ) |` om ) Fn om )
16		fveq2 5627	. . . . 5 |- ( x = (/) -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` (/) ) )
17		fveq2 5627	. . . . 5 |- ( x = (/) -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` (/) ) )
18	16, 17	eqeq12d 2244	. . . 4 |- ( x = (/) -> ( (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) <-> (frec ( F , A ) ` (/) ) = ( ( rec ( F , A ) |` om ) ` (/) ) ) )
19		fveq2 5627	. . . . 5 |- ( x = y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` y ) )
20		fveq2 5627	. . . . 5 |- ( x = y -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` y ) )
21	19, 20	eqeq12d 2244	. . . 4 |- ( x = y -> ( (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) <-> (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) )
22		fveq2 5627	. . . . 5 |- ( x = suc y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` suc y ) )
23		fveq2 5627	. . . . 5 |- ( x = suc y -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` suc y ) )
24	22, 23	eqeq12d 2244	. . . 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 6543	. . . . . 6 |- ( A e. V -> (frec ( F , A ) ` (/) ) = A )
26	8, 25	syl 14	. . . . 5 |- ( ph -> (frec ( F , A ) ` (/) ) = A )
27		peano1 4686	. . . . . . 7 |- (/) e. om
28		fvres 5651	. . . . . . 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 6534	. . . . . . 7 |- ( A e. V -> ( rec ( F , A ) ` (/) ) = A )
31	8, 30	syl 14	. . . . . 6 |- ( ph -> ( rec ( F , A ) ` (/) ) = A )
32	29, 31	eqtrid 2274	. . . . 5 |- ( ph -> ( ( rec ( F , A ) |` om ) ` (/) ) = A )
33	26, 32	eqtr4d 2265	. . . 4 |- ( ph -> (frec ( F , A ) ` (/) ) = ( ( rec ( F , A ) |` om ) ` (/) ) )
34		simpr 110	. . . . . . . . . 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 5651	. . . . . . . . . . 11 |- ( y e. om -> ( ( rec ( F , A ) |` om ) ` y ) = ( rec ( F , A ) ` y ) )
36	35	ad2antlr 489	. . . . . . . . . 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 2262	. . . . . . . . 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 5631	. . . . . . . 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 306	. . . . . . . . . 10 |- ( ph -> ( A. z ( F ` z ) e. _V /\ A e. V ) )
40		simp1 1021	. . . . . . . . . . . . 13 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A. z ( F ` z ) e. _V )
41		ralv 2817	. . . . . . . . . . . . 13 |- ( A. z e. _V ( F ` z ) e. _V <-> A. z ( F ` z ) e. _V )
42	40, 41	sylibr 134	. . . . . . . . . . . 12 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A. z e. _V ( F ` z ) e. _V )
43		simp2 1022	. . . . . . . . . . . . 13 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A e. V )
44	43	elexd 2813	. . . . . . . . . . . 12 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A e. _V )
45		simp3 1023	. . . . . . . . . . . 12 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> y e. om )
46		frecsuc 6553	. . . . . . . . . . . 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 1271	. . . . . . . . . . 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 1227	. . . . . . . . . 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 283	. . . . . . . . 9 |- ( ( ph /\ y e. om ) -> (frec ( F , A ) ` suc y ) = ( F ` (frec ( F , A ) ` y ) ) )
50	49	adantr 276	. . . . . . . 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 276	. . . . . . . . . 10 |- ( ( ph /\ y e. om ) -> F Fn _V )
52	8	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ y e. om ) -> A e. V )
53		simpr 110	. . . . . . . . . . 11 |- ( ( ph /\ y e. om ) -> y e. om )
54		nnon 4702	. . . . . . . . . . 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 276	. . . . . . . . . 10 |- ( ( ph /\ y e. om ) -> A. x x C_ ( F ` x ) )
58	51, 52, 55, 57	rdgisucinc 6531	. . . . . . . . 9 |- ( ( ph /\ y e. om ) -> ( rec ( F , A ) ` suc y ) = ( F ` ( rec ( F , A ) ` y ) ) )
59	58	adantr 276	. . . . . . . 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 2272	. . . . . . 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 4687	. . . . . . . . 9 |- ( y e. om -> suc y e. om )
62		fvres 5651	. . . . . . . . 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 489	. . . . . . 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 2265	. . . . . 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 115	. . . . 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 116	. . . 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 4693	. . 3 |- ( x e. om -> ( ph -> (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) ) )
69	68	impcom 125	. 2 |- ( ( ph /\ x e. om ) -> (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) )
70	10, 15, 69	eqfnfvd 5735	1 |- ( ph -> frec ( F , A ) = ( rec ( F , A ) |` om ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   A.wal 1393   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   C_ wss 3197   (/)c0 3491   Oncon0 4454   suc csuc 4456   omcom 4682   |` cres 4721   Fn wfn 5313   ` cfv 5318   reccrdg 6515  freccfrec 6536

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-recs 6451  df-irdg 6516  df-frec 6537

This theorem is referenced by: (None)