Theorem frecrdg 6411

Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6394 produces the same results as df-irdg 6373 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 2742	. . . . . 6 |- z e. _V
3		funfvex 5534	. . . . . . 7 |- ( ( Fun F /\ z e. dom F ) -> ( F ` z ) e. _V )
4	3	funfni 5318	. . . . . 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 1874	. . . 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 6404	. . 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 6383	. . . 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 4614	. . 3 |- om C_ On
14		fnssres 5331	. . 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 5517	. . . . 5 |- ( x = (/) -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` (/) ) )
17		fveq2 5517	. . . . 5 |- ( x = (/) -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` (/) ) )
18	16, 17	eqeq12d 2192	. . . 4 |- ( x = (/) -> ( (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) <-> (frec ( F , A ) ` (/) ) = ( ( rec ( F , A ) |` om ) ` (/) ) ) )
19		fveq2 5517	. . . . 5 |- ( x = y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` y ) )
20		fveq2 5517	. . . . 5 |- ( x = y -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` y ) )
21	19, 20	eqeq12d 2192	. . . 4 |- ( x = y -> ( (frec ( F , A ) ` x ) = ( ( rec ( F , A ) |` om ) ` x ) <-> (frec ( F , A ) ` y ) = ( ( rec ( F , A ) |` om ) ` y ) ) )
22		fveq2 5517	. . . . 5 |- ( x = suc y -> (frec ( F , A ) ` x ) = (frec ( F , A ) ` suc y ) )
23		fveq2 5517	. . . . 5 |- ( x = suc y -> ( ( rec ( F , A ) |` om ) ` x ) = ( ( rec ( F , A ) |` om ) ` suc y ) )
24	22, 23	eqeq12d 2192	. . . 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 6400	. . . . . 6 |- ( A e. V -> (frec ( F , A ) ` (/) ) = A )
26	8, 25	syl 14	. . . . 5 |- ( ph -> (frec ( F , A ) ` (/) ) = A )
27		peano1 4595	. . . . . . 7 |- (/) e. om
28		fvres 5541	. . . . . . 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 6391	. . . . . . 7 |- ( A e. V -> ( rec ( F , A ) ` (/) ) = A )
31	8, 30	syl 14	. . . . . 6 |- ( ph -> ( rec ( F , A ) ` (/) ) = A )
32	29, 31	eqtrid 2222	. . . . 5 |- ( ph -> ( ( rec ( F , A ) |` om ) ` (/) ) = A )
33	26, 32	eqtr4d 2213	. . . 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 5541	. . . . . . . . . . 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 2210	. . . . . . . . 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 5521	. . . . . . . 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 997	. . . . . . . . . . . . 13 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A. z ( F ` z ) e. _V )
41		ralv 2756	. . . . . . . . . . . . 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 998	. . . . . . . . . . . . 13 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A e. V )
44	43	elexd 2752	. . . . . . . . . . . 12 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> A e. _V )
45		simp3 999	. . . . . . . . . . . 12 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ y e. om ) -> y e. om )
46		frecsuc 6410	. . . . . . . . . . . 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 1238	. . . . . . . . . . 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 1203	. . . . . . . . . 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 4611	. . . . . . . . . . 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 6388	. . . . . . . . 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 2220	. . . . . . 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 4596	. . . . . . . . 9 |- ( y e. om -> suc y e. om )
62		fvres 5541	. . . . . . . . 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 2213	. . . . . 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 4602	. . 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 5618	1 |- ( ph -> frec ( F , A ) = ( rec ( F , A ) |` om ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   A.wal 1351   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2739   C_ wss 3131   (/)c0 3424   Oncon0 4365   suc csuc 4367   omcom 4591   |` cres 4630   Fn wfn 5213   ` cfv 5218   reccrdg 6372  freccfrec 6393

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-recs 6308  df-irdg 6373  df-frec 6394

This theorem is referenced by: (None)