Theorem tfrcllemubacc 6379

Description: Lemma for tfrcl 6384. The union of B satisfies the recursion rule. (Contributed by Jim Kingdon, 25-Mar-2022.)

Hypotheses

Ref	Expression
tfrcl.f	|- F = recs ( G )
tfrcl.g	|- ( ph -> Fun G )
tfrcl.x	|- ( ph -> Ord X )
tfrcl.ex	|- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )
tfrcllemsucfn.1	|- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
tfrcllembacc.3	|- B = { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }
tfrcllembacc.u	|- ( ( ph /\ x e. U. X ) -> suc x e. X )
tfrcllembacc.4	|- ( ph -> D e. X )
tfrcllembacc.5	|- ( ph -> A. z e. D E. g ( g : z --> S /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )

Assertion

Ref	Expression
tfrcllemubacc	|- ( ph -> A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) )

Distinct variable groups:   A, f, g, h, x, y, z   D, f, g, x, y   f, G, x, y   S, f, x, y   f, X, x   ph, f, g, h, x, y, z   B, g, h, z   u, B, w   D, h, z   u, D, w   w, G   h, G, z   u, G   S, g, h, z   z, X   w, g, ph, y, z

Allowed substitution hints:   ph( u)   A( w, u)   B( x, y, f)   S( w, u)   F( x, y, z, w, u, f, g, h)   G( g)   X( y, w, u, g, h)

Proof of Theorem tfrcllemubacc

Dummy variables e t v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrcl.f	. . . . . . . . 9 |- F = recs ( G )
2		tfrcl.g	. . . . . . . . 9 |- ( ph -> Fun G )
3		tfrcl.x	. . . . . . . . 9 |- ( ph -> Ord X )
4		tfrcl.ex	. . . . . . . . 9 |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )
5		tfrcllemsucfn.1	. . . . . . . . 9 |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
6		tfrcllembacc.3	. . . . . . . . 9 |- B = { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }
7		tfrcllembacc.u	. . . . . . . . 9 |- ( ( ph /\ x e. U. X ) -> suc x e. X )
8		tfrcllembacc.4	. . . . . . . . 9 |- ( ph -> D e. X )
9		tfrcllembacc.5	. . . . . . . . 9 |- ( ph -> A. z e. D E. g ( g : z --> S /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	tfrcllembfn 6377	. . . . . . . 8 |- ( ph -> U. B : D --> S )
11		fdm 5387	. . . . . . . 8 |- ( U. B : D --> S -> dom U. B = D )
12	10, 11	syl 14	. . . . . . 7 |- ( ph -> dom U. B = D )
13	1, 2, 3, 4, 5, 6, 7, 8, 9	tfrcllembacc 6375	. . . . . . . . . 10 |- ( ph -> B C_ A )
14	13	unissd 3848	. . . . . . . . 9 |- ( ph -> U. B C_ U. A )
15	5, 3	tfrcllemssrecs 6372	. . . . . . . . 9 |- ( ph -> U. A C_ recs ( G ) )
16	14, 15	sstrd 3180	. . . . . . . 8 |- ( ph -> U. B C_ recs ( G ) )
17		dmss 4841	. . . . . . . 8 |- ( U. B C_ recs ( G ) -> dom U. B C_ dom recs ( G ) )
18	16, 17	syl 14	. . . . . . 7 |- ( ph -> dom U. B C_ dom recs ( G ) )
19	12, 18	eqsstrrd 3207	. . . . . 6 |- ( ph -> D C_ dom recs ( G ) )
20	19	sselda 3170	. . . . 5 |- ( ( ph /\ w e. D ) -> w e. dom recs ( G ) )
21		eqid 2189	. . . . . 6 |- { e | E. v e. On ( e Fn v /\ A. t e. v ( e ` t ) = ( G ` ( e |` t ) ) ) } = { e | E. v e. On ( e Fn v /\ A. t e. v ( e ` t ) = ( G ` ( e |` t ) ) ) }
22	21	tfrlem9 6339	. . . . 5 |- ( w e. dom recs ( G ) -> (recs ( G ) ` w ) = ( G ` (recs ( G ) |` w ) ) )
23	20, 22	syl 14	. . . 4 |- ( ( ph /\ w e. D ) -> (recs ( G ) ` w ) = ( G ` (recs ( G ) |` w ) ) )
24		tfrfun 6340	. . . . 5 |- Fun recs ( G )
25	12	eleq2d 2259	. . . . . 6 |- ( ph -> ( w e. dom U. B <-> w e. D ) )
26	25	biimpar 297	. . . . 5 |- ( ( ph /\ w e. D ) -> w e. dom U. B )
27		funssfv 5557	. . . . 5 |- ( ( Fun recs ( G ) /\ U. B C_ recs ( G ) /\ w e. dom U. B ) -> (recs ( G ) ` w ) = ( U. B ` w ) )
28	24, 16, 26, 27	mp3an2ani 1355	. . . 4 |- ( ( ph /\ w e. D ) -> (recs ( G ) ` w ) = ( U. B ` w ) )
29		ordelon 4398	. . . . . . . . . 10 |- ( ( Ord X /\ D e. X ) -> D e. On )
30	3, 8, 29	syl2anc 411	. . . . . . . . 9 |- ( ph -> D e. On )
31		eloni 4390	. . . . . . . . 9 |- ( D e. On -> Ord D )
32	30, 31	syl 14	. . . . . . . 8 |- ( ph -> Ord D )
33		ordelss 4394	. . . . . . . 8 |- ( ( Ord D /\ w e. D ) -> w C_ D )
34	32, 33	sylan 283	. . . . . . 7 |- ( ( ph /\ w e. D ) -> w C_ D )
35	12	adantr 276	. . . . . . 7 |- ( ( ph /\ w e. D ) -> dom U. B = D )
36	34, 35	sseqtrrd 3209	. . . . . 6 |- ( ( ph /\ w e. D ) -> w C_ dom U. B )
37		fun2ssres 5275	. . . . . 6 |- ( ( Fun recs ( G ) /\ U. B C_ recs ( G ) /\ w C_ dom U. B ) -> (recs ( G ) |` w ) = ( U. B |` w ) )
38	24, 16, 36, 37	mp3an2ani 1355	. . . . 5 |- ( ( ph /\ w e. D ) -> (recs ( G ) |` w ) = ( U. B |` w ) )
39	38	fveq2d 5535	. . . 4 |- ( ( ph /\ w e. D ) -> ( G ` (recs ( G ) |` w ) ) = ( G ` ( U. B |` w ) ) )
40	23, 28, 39	3eqtr3d 2230	. . 3 |- ( ( ph /\ w e. D ) -> ( U. B ` w ) = ( G ` ( U. B |` w ) ) )
41	40	ralrimiva 2563	. 2 |- ( ph -> A. w e. D ( U. B ` w ) = ( G ` ( U. B |` w ) ) )
42		fveq2 5531	. . . 4 |- ( u = w -> ( U. B ` u ) = ( U. B ` w ) )
43		reseq2 4917	. . . . 5 |- ( u = w -> ( U. B |` u ) = ( U. B |` w ) )
44	43	fveq2d 5535	. . . 4 |- ( u = w -> ( G ` ( U. B |` u ) ) = ( G ` ( U. B |` w ) ) )
45	42, 44	eqeq12d 2204	. . 3 |- ( u = w -> ( ( U. B ` u ) = ( G ` ( U. B |` u ) ) <-> ( U. B ` w ) = ( G ` ( U. B |` w ) ) ) )
46	45	cbvralv 2718	. 2 |- ( A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) <-> A. w e. D ( U. B ` w ) = ( G ` ( U. B |` w ) ) )
47	41, 46	sylibr 134	1 |- ( ph -> A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469   u. cun 3142   C_ wss 3144   {csn 3607   <.cop 3610   U.cuni 3824   Ord word 4377   Oncon0 4378   suc csuc 4380   dom cdm 4641   |` cres 4643   Fun wfun 5226   Fn wfn 5227   -->wf 5228   ` cfv 5232  recscrecs 6324

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-recs 6325

This theorem is referenced by:  tfrcllemex  6380