Theorem frecrdg 6387

Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6370 produces the same results as df-irdg 6349 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 2733	. . . . . 6 |- z e. _V
3		funfvex 5513	. . . . . . 7 |- ( ( Fun F /\ z e. dom F ) -> ( F ` z ) e. _V )
4	3	funfni 5298	. . . . . 6 |- ( ( F Fn _V /\ z e. _V ) -> ( F ` z ) e. _V )
5	2, 4	mpan2 423	. . . . 5 |- ( F Fn _V -> ( F ` z ) e. _V )
6	5	alrimiv 1867	. . . 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 6380	. . 3 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> frec ( F , A ) Fn om )
10	7, 8, 9	syl2anc 409	. 2 |- ( ph -> frec ( F , A ) Fn om )
11		rdgifnon2 6359	. . . 4 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> rec ( F , A ) Fn On )
12	7, 8, 11	syl2anc 409	. . 3 |- ( ph -> rec ( F , A ) Fn On )
13		omsson 4597	. . 3 |- om C_ On
14		fnssres 5311	. . 3 |- ( ( rec ( F , A ) Fn On /\ om C_ On ) -> ( rec ( F , A ) |` om ) Fn om )
15	12, 13, 14	sylancl 411	. 2 |- ( ph -> ( rec ( F , A ) |` om ) Fn om )
16		fveq2 5496	. . . . 5 |- ( x = (/) -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` (/) ) )
17		fveq2 5496	. . . . 5 |- ( x = (/) -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` (/) ) )
18	16, 17	eqeq12d 2185	. . . 4 |- ( x = (/) -> ( (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) <-> (frec ( F , A ) ` (/) ) = ( ( rec ( F , A ) |` om ) ` (/) ) ) )
19		fveq2 5496	. . . . 5 |- ( x = y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` y ) )
20		fveq2 5496	. . . . 5 |- ( x = y -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` y ) )
21	19, 20	eqeq12d 2185	. . . 4 |- ( x = y -> ( (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) <-> (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) )
22		fveq2 5496	. . . . 5 |- ( x = suc y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` suc y ) )
23		fveq2 5496	. . . . 5 |- ( x = suc y -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` suc y ) )
24	22, 23	eqeq12d 2185	. . . 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 6376	. . . . . 6 |- ( A e. V -> (frec ( F , A ) ` (/) ) = A )
26	8, 25	syl 14	. . . . 5 |- ( ph -> (frec ( F , A ) ` (/) ) = A )
27		peano1 4578	. . . . . . 7 |- (/) e. om
28		fvres 5520	. . . . . . 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 6367	. . . . . . 7 |- ( A e. V -> ( rec ( F , A ) ` (/) ) = A )
31	8, 30	syl 14	. . . . . 6 |- ( ph -> ( rec ( F , A ) ` (/) ) = A )
32	29, 31	eqtrid 2215	. . . . 5 |- ( ph -> ( ( rec ( F , A ) |` om ) ` (/) ) = A )
33	26, 32	eqtr4d 2206	. . . 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 5520	. . . . . . . . . . 11 |- ( y e. om -> ( ( rec ( F , A ) |` om ) ` y ) = ( rec ( F , A ) ` y ) )
36	35	ad2antlr 486	. . . . . . . . . 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 2203	. . . . . . . . 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 5500	. . . . . . . 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 992	. . . . . . . . . . . . 13 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A. z ( F ` z ) e. _V )
41		ralv 2747	. . . . . . . . . . . . 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 993	. . . . . . . . . . . . 13 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A e. V )
44	43	elexd 2743	. . . . . . . . . . . 12 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A e. _V )
45		simp3 994	. . . . . . . . . . . 12 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> y e. om )
46		frecsuc 6386	. . . . . . . . . . . 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 1233	. . . . . . . . . . 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 1198	. . . . . . . . . 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 4594	. . . . . . . . . . 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 6364	. . . . . . . . 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 2213	. . . . . . 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 4579	. . . . . . . . 9 |- ( y e. om -> suc y e. om )
62		fvres 5520	. . . . . . . . 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 486	. . . . . . 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 2206	. . . . . 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 4585	. . 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 5596	1 |- ( ph -> frec ( F , A ) = ( rec ( F , A ) |` om ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   A.wal 1346   = wceq 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   C_ wss 3121   (/)c0 3414   Oncon0 4348   suc csuc 4350   omcom 4574   |` cres 4613   Fn wfn 5193   ` cfv 5198   reccrdg 6348  freccfrec 6369

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-recs 6284  df-irdg 6349  df-frec 6370

This theorem is referenced by: (None)