Theorem rdgon 5996

Description: Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.)

Hypotheses

Ref	Expression
rdgon.1	|- ( ph -> F Fn _V )
rdgon.2	|- ( ph -> A e. On )
rdgon.3	|- ( ph -> A. x e. On ( F ` x ) e. On )

Assertion

Ref	Expression
rdgon	|- ( ( ph /\ B e. On ) -> ( rec ( F , A ) ` B ) e. On )

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

Allowed substitution hint:   B( x)

Proof of Theorem rdgon

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

Step	Hyp	Ref	Expression
1		fveq2 5198	. . . . 5 |- ( z = x -> ( rec ( F , A ) ` z ) = ( rec ( F , A ) ` x ) )
2	1	eleq1d 2147	. . . 4 |- ( z = x -> ( ( rec ( F , A ) ` z ) e. On <-> ( rec ( F , A ) ` x ) e. On ) )
3	2	imbi2d 228	. . 3 |- ( z = x -> ( ( ph -> ( rec ( F , A ) ` z ) e. On ) <-> ( ph -> ( rec ( F , A ) ` x ) e. On ) ) )
4		fveq2 5198	. . . . 5 |- ( z = B -> ( rec ( F , A ) ` z ) = ( rec ( F , A ) ` B ) )
5	4	eleq1d 2147	. . . 4 |- ( z = B -> ( ( rec ( F , A ) ` z ) e. On <-> ( rec ( F , A ) ` B ) e. On ) )
6	5	imbi2d 228	. . 3 |- ( z = B -> ( ( ph -> ( rec ( F , A ) ` z ) e. On ) <-> ( ph -> ( rec ( F , A ) ` B ) e. On ) ) )
7		r19.21v 2438	. . . 4 |- ( A. x e. z ( ph -> ( rec ( F , A ) ` x ) e. On ) <-> ( ph -> A. x e. z ( rec ( F , A ) ` x ) e. On ) )
8		rdgon.2	. . . . . . . . 9 |- ( ph -> A e. On )
9		fvres 5219	. . . . . . . . . . . . . 14 |- ( x e. z -> ( ( rec ( F , A ) |` z ) ` x ) = ( rec ( F , A ) ` x ) )
10	9	eleq1d 2147	. . . . . . . . . . . . 13 |- ( x e. z -> ( ( ( rec ( F , A ) |` z ) ` x ) e. On <-> ( rec ( F , A ) ` x ) e. On ) )
11	10	adantl 271	. . . . . . . . . . . 12 |- ( ( ph /\ x e. z ) -> ( ( ( rec ( F , A ) |` z ) ` x ) e. On <-> ( rec ( F , A ) ` x ) e. On ) )
12		rdgon.3	. . . . . . . . . . . . . . 15 |- ( ph -> A. x e. On ( F ` x ) e. On )
13		fveq2 5198	. . . . . . . . . . . . . . . . 17 |- ( x = w -> ( F ` x ) = ( F ` w ) )
14	13	eleq1d 2147	. . . . . . . . . . . . . . . 16 |- ( x = w -> ( ( F ` x ) e. On <-> ( F ` w ) e. On ) )
15	14	cbvralv 2577	. . . . . . . . . . . . . . 15 |- ( A. x e. On ( F ` x ) e. On <-> A. w e. On ( F ` w ) e. On )
16	12, 15	sylib 120	. . . . . . . . . . . . . 14 |- ( ph -> A. w e. On ( F ` w ) e. On )
17		fveq2 5198	. . . . . . . . . . . . . . . 16 |- ( w = ( ( rec ( F , A ) |` z ) ` x ) -> ( F ` w ) = ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) )
18	17	eleq1d 2147	. . . . . . . . . . . . . . 15 |- ( w = ( ( rec ( F , A ) |` z ) ` x ) -> ( ( F ` w ) e. On <-> ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) e. On ) )
19	18	rspcv 2697	. . . . . . . . . . . . . 14 |- ( ( ( rec ( F , A ) |` z ) ` x ) e. On -> ( A. w e. On ( F ` w ) e. On -> ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) e. On ) )
20	16, 19	syl5com 29	. . . . . . . . . . . . 13 |- ( ph -> ( ( ( rec ( F , A ) |` z ) ` x ) e. On -> ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) e. On ) )
21	20	adantr 270	. . . . . . . . . . . 12 |- ( ( ph /\ x e. z ) -> ( ( ( rec ( F , A ) |` z ) ` x ) e. On -> ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) e. On ) )
22	11, 21	sylbird 168	. . . . . . . . . . 11 |- ( ( ph /\ x e. z ) -> ( ( rec ( F , A ) ` x ) e. On -> ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) e. On ) )
23	22	ralimdva 2429	. . . . . . . . . 10 |- ( ph -> ( A. x e. z ( rec ( F , A ) ` x ) e. On -> A. x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) e. On ) )
24		vex 2604	. . . . . . . . . . 11 |- z e. _V
25		iunon 5922	. . . . . . . . . . 11 |- ( ( z e. _V /\ A. x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) e. On ) -> U_ x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) e. On )
26	24, 25	mpan 414	. . . . . . . . . 10 |- ( A. x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) e. On -> U_ x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) e. On )
27	23, 26	syl6 33	. . . . . . . . 9 |- ( ph -> ( A. x e. z ( rec ( F , A ) ` x ) e. On -> U_ x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) e. On ) )
28		onun2 4234	. . . . . . . . 9 |- ( ( A e. On /\ U_ x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) e. On ) -> ( A u. U_ x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) ) e. On )
29	8, 27, 28	syl6an 1363	. . . . . . . 8 |- ( ph -> ( A. x e. z ( rec ( F , A ) ` x ) e. On -> ( A u. U_ x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) ) e. On ) )
30	29	adantr 270	. . . . . . 7 |- ( ( ph /\ z e. On ) -> ( A. x e. z ( rec ( F , A ) ` x ) e. On -> ( A u. U_ x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) ) e. On ) )
31		rdgon.1	. . . . . . . . . 10 |- ( ph -> F Fn _V )
32	31, 8	jca 300	. . . . . . . . 9 |- ( ph -> ( F Fn _V /\ A e. On ) )
33		rdgivallem 5991	. . . . . . . . . 10 |- ( ( F Fn _V /\ A e. On /\ z e. On ) -> ( rec ( F , A ) ` z ) = ( A u. U_ x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) ) )
34	33	3expa 1138	. . . . . . . . 9 |- ( ( ( F Fn _V /\ A e. On ) /\ z e. On ) -> ( rec ( F , A ) ` z ) = ( A u. U_ x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) ) )
35	32, 34	sylan 277	. . . . . . . 8 |- ( ( ph /\ z e. On ) -> ( rec ( F , A ) ` z ) = ( A u. U_ x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) ) )
36	35	eleq1d 2147	. . . . . . 7 |- ( ( ph /\ z e. On ) -> ( ( rec ( F , A ) ` z ) e. On <-> ( A u. U_ x e. z ( F ` ( ( rec ( F , A ) |` z ) ` x ) ) ) e. On ) )
37	30, 36	sylibrd 167	. . . . . 6 |- ( ( ph /\ z e. On ) -> ( A. x e. z ( rec ( F , A ) ` x ) e. On -> ( rec ( F , A ) ` z ) e. On ) )
38	37	expcom 114	. . . . 5 |- ( z e. On -> ( ph -> ( A. x e. z ( rec ( F , A ) ` x ) e. On -> ( rec ( F , A ) ` z ) e. On ) ) )
39	38	a2d 26	. . . 4 |- ( z e. On -> ( ( ph -> A. x e. z ( rec ( F , A ) ` x ) e. On ) -> ( ph -> ( rec ( F , A ) ` z ) e. On ) ) )
40	7, 39	syl5bi 150	. . 3 |- ( z e. On -> ( A. x e. z ( ph -> ( rec ( F , A ) ` x ) e. On ) -> ( ph -> ( rec ( F , A ) ` z ) e. On ) ) )
41	3, 6, 40	tfis3 4327	. 2 |- ( B e. On -> ( ph -> ( rec ( F , A ) ` B ) e. On ) )
42	41	impcom 123	1 |- ( ( ph /\ B e. On ) -> ( rec ( F , A ) ` B ) e. On )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   _Vcvv 2601   u. cun 2971   U_ciun 3678   Oncon0 4118   |` cres 4365   Fn wfn 4917   ` cfv 4922   reccrdg 5979

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-recs 5943  df-irdg 5980

This theorem is referenced by:  oacl  6063  omcl  6064  oeicl  6065