Theorem tfr1onlemubacc 6404

Description: Lemma for tfr1on 6408. 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 6402	. . . . . . . 8 |- ( ph -> U. B Fn D )
11		fndm 5357	. . . . . . . 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 6400	. . . . . . . . . 10 |- ( ph -> B C_ A )
14	13	unissd 3863	. . . . . . . . 9 |- ( ph -> U. B C_ U. A )
15	5, 3	tfr1onlemssrecs 6397	. . . . . . . . 9 |- ( ph -> U. A C_ recs ( G ) )
16	14, 15	sstrd 3193	. . . . . . . 8 |- ( ph -> U. B C_ recs ( G ) )
17		dmss 4865	. . . . . . . 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 3220	. . . . . 6 |- ( ph -> D C_ dom recs ( G ) )
20	19	sselda 3183	. . . . 5 |- ( ( ph /\ w e. D ) -> w e. dom recs ( G ) )
21		eqid 2196	. . . . . 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 6377	. . . . 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 6378	. . . . 5 |- Fun recs ( G )
25	12	eleq2d 2266	. . . . . 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 5584	. . . . 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 4418	. . . . . . . . . 10 |- ( ( Ord X /\ D e. X ) -> D e. On )
30	3, 8, 29	syl2anc 411	. . . . . . . . 9 |- ( ph -> D e. On )
31		eloni 4410	. . . . . . . . 9 |- ( D e. On -> Ord D )
32	30, 31	syl 14	. . . . . . . 8 |- ( ph -> Ord D )
33		ordelss 4414	. . . . . . . 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 3222	. . . . . 6 |- ( ( ph /\ w e. D ) -> w C_ dom U. B )
37		fun2ssres 5301	. . . . . 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 5562	. . . 4 |- ( ( ph /\ w e. D ) -> ( G ` (recs ( G ) |` w ) ) = ( G ` ( U. B |` w ) ) )
40	23, 28, 39	3eqtr3d 2237	. . 3 |- ( ( ph /\ w e. D ) -> ( U. B ` w ) = ( G ` ( U. B |` w ) ) )
41	40	ralrimiva 2570	. 2 |- ( ph -> A. w e. D ( U. B ` w ) = ( G ` ( U. B |` w ) ) )
42		fveq2 5558	. . . 4 |- ( u = w -> ( U. B ` u ) = ( U. B ` w ) )
43		reseq2 4941	. . . . 5 |- ( u = w -> ( U. B |` u ) = ( U. B |` w ) )
44	43	fveq2d 5562	. . . 4 |- ( u = w -> ( G ` ( U. B |` u ) ) = ( G ` ( U. B |` w ) ) )
45	42, 44	eqeq12d 2211	. . 3 |- ( u = w -> ( ( U. B ` u ) = ( G ` ( U. B |` u ) ) <-> ( U. B ` w ) = ( G ` ( U. B |` w ) ) ) )
46	45	cbvralv 2729	. 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 1506   e. wcel 2167   {cab 2182   A.wral 2475   E.wrex 2476   _Vcvv 2763   u. cun 3155   C_ wss 3157   {csn 3622   <.cop 3625   U.cuni 3839   Ord word 4397   Oncon0 4398   suc csuc 4400   dom cdm 4663   |` cres 4665   Fun wfun 5252   Fn wfn 5253   ` cfv 5258  recscrecs 6362

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-recs 6363

This theorem is referenced by:  tfr1onlemex  6405