Theorem tfr1onlemubacc 6015

Description: Lemma for tfr1on 6019. The union of B satisfies the recursion rule. (Contributed by Jim Kingdon, 15-Mar-2022.)

Hypotheses

Ref	Expression
tfr1on.f	|- F = recs ( G )
tfr1on.g	|- ( ph -> Fun G )
tfr1on.x	|- ( ph -> Ord X )
tfr1on.ex	|- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
tfr1onlemsucfn.1	|- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
tfr1onlembacc.3	|- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }
tfr1onlembacc.u	|- ( ( ph /\ x e. U. X ) -> suc x e. X )
tfr1onlembacc.4	|- ( ph -> D e. X )
tfr1onlembacc.5	|- ( ph -> A. z e. D E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )

Assertion

Ref	Expression
tfr1onlemubacc	|- ( ph -> A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) )

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

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

Proof of Theorem tfr1onlemubacc

Step	Hyp	Ref	Expression
1		tfr1on.f	. . . . . . . . 9 |- F = recs ( G )
2		tfr1on.g	. . . . . . . . 9 |- ( ph -> Fun G )
3		tfr1on.x	. . . . . . . . 9 |- ( ph -> Ord X )
4		tfr1on.ex	. . . . . . . . 9 |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )
5		tfr1onlemsucfn.1	. . . . . . . . 9 |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
6		tfr1onlembacc.3	. . . . . . . . 9 |- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }
7		tfr1onlembacc.u	. . . . . . . . 9 |- ( ( ph /\ x e. U. X ) -> suc x e. X )
8		tfr1onlembacc.4	. . . . . . . . 9 |- ( ph -> D e. X )
9		tfr1onlembacc.5	. . . . . . . . 9 |- ( ph -> A. z e. D E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlembfn 6013	. . . . . . . 8 |- ( ph -> U. B Fn D )
11		fndm 5049	. . . . . . . 8 |- ( U. B Fn D -> 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	tfr1onlembacc 6011	. . . . . . . . . 10 |- ( ph -> B C_ A )
14	13	unissd 3645	. . . . . . . . 9 |- ( ph -> U. B C_ U. A )
15	5, 3	tfr1onlemssrecs 6008	. . . . . . . . 9 |- ( ph -> U. A C_ recs ( G ) )
16	14, 15	sstrd 3018	. . . . . . . 8 |- ( ph -> U. B C_ recs ( G ) )
17		dmss 4582	. . . . . . . 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	eqsstr3d 3043	. . . . . 6 |- ( ph -> D C_ dom recs ( G ) )
20	19	sselda 3008	. . . . 5 |- ( ( ph /\ w e. D ) -> w e. dom recs ( G ) )
21		eqid 2083	. . . . . 6 |- { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
22	21	tfrlem9 5988	. . . . 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 5989	. . . . 5 |- Fun recs ( G )
25	12	eleq2d 2152	. . . . . 6 |- ( ph -> ( w e. dom U. B <-> w e. D ) )
26	25	biimpar 291	. . . . 5 |- ( ( ph /\ w e. D ) -> w e. dom U. B )
27		funssfv 5251	. . . . 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 1276	. . . 4 |- ( ( ph /\ w e. D ) -> (recs ( G ) ` w ) = ( U. B ` w ) )
29		ordelon 4166	. . . . . . . . . 10 |- ( ( Ord X /\ D e. X ) -> D e. On )
30	3, 8, 29	syl2anc 403	. . . . . . . . 9 |- ( ph -> D e. On )
31		eloni 4158	. . . . . . . . 9 |- ( D e. On -> Ord D )
32	30, 31	syl 14	. . . . . . . 8 |- ( ph -> Ord D )
33		ordelss 4162	. . . . . . . 8 |- ( ( Ord D /\ w e. D ) -> w C_ D )
34	32, 33	sylan 277	. . . . . . 7 |- ( ( ph /\ w e. D ) -> w C_ D )
35	12	adantr 270	. . . . . . 7 |- ( ( ph /\ w e. D ) -> dom U. B = D )
36	34, 35	sseqtr4d 3045	. . . . . 6 |- ( ( ph /\ w e. D ) -> w C_ dom U. B )
37		fun2ssres 4993	. . . . . 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 1276	. . . . 5 |- ( ( ph /\ w e. D ) -> (recs ( G ) |` w ) = ( U. B |` w ) )
39	38	fveq2d 5233	. . . 4 |- ( ( ph /\ w e. D ) -> ( G ` (recs ( G ) |` w ) ) = ( G ` ( U. B |` w ) ) )
40	23, 28, 39	3eqtr3d 2123	. . 3 |- ( ( ph /\ w e. D ) -> ( U. B ` w ) = ( G ` ( U. B |` w ) ) )
41	40	ralrimiva 2439	. 2 |- ( ph -> A. w e. D ( U. B ` w ) = ( G ` ( U. B |` w ) ) )
42		fveq2 5229	. . . 4 |- ( u = w -> ( U. B ` u ) = ( U. B ` w ) )
43		reseq2 4655	. . . . 5 |- ( u = w -> ( U. B |` u ) = ( U. B |` w ) )
44	43	fveq2d 5233	. . . 4 |- ( u = w -> ( G ` ( U. B |` u ) ) = ( G ` ( U. B |` w ) ) )
45	42, 44	eqeq12d 2097	. . 3 |- ( u = w -> ( ( U. B ` u ) = ( G ` ( U. B |` u ) ) <-> ( U. B ` w ) = ( G ` ( U. B |` w ) ) ) )
46	45	cbvralv 2582	. 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 132	1 |- ( ph -> A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 920   = wceq 1285   E.wex 1422   e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   _Vcvv 2610   u. cun 2980   C_ wss 2982   {csn 3416   <.cop 3419   U.cuni 3621   Ord word 4145   Oncon0 4146   suc csuc 4148   dom cdm 4391   |` cres 4393   Fun wfun 4946   Fn wfn 4947   ` cfv 4952  recscrecs 5973

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960  df-recs 5974

This theorem is referenced by:  tfr1onlemex  6016