Theorem tfrlemiubacc 6298

Description: The union of B satisfies the recursion rule (lemma for tfrlemi1 6300). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Hypotheses

Ref	Expression
tfrlemisucfn.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
tfrlemisucfn.2	|- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )
tfrlemi1.3	|- B = { h | E. z e. x E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) }
tfrlemi1.4	|- ( ph -> x e. On )
tfrlemi1.5	|- ( ph -> A. z e. x E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )

Assertion

Ref	Expression
tfrlemiubacc	|- ( ph -> A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) )

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

Allowed substitution hints:   ph( x, u, f)   B( x, y)

Proof of Theorem tfrlemiubacc

Step	Hyp	Ref	Expression
1		tfrlemisucfn.1	. . . . . . . . 9 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
2		tfrlemisucfn.2	. . . . . . . . 9 |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )
3		tfrlemi1.3	. . . . . . . . 9 |- B = { h | E. z e. x E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) }
4		tfrlemi1.4	. . . . . . . . 9 |- ( ph -> x e. On )
5		tfrlemi1.5	. . . . . . . . 9 |- ( ph -> A. z e. x E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )
6	1, 2, 3, 4, 5	tfrlemibfn 6296	. . . . . . . 8 |- ( ph -> U. B Fn x )
7		fndm 5287	. . . . . . . 8 |- ( U. B Fn x -> dom U. B = x )
8	6, 7	syl 14	. . . . . . 7 |- ( ph -> dom U. B = x )
9	1, 2, 3, 4, 5	tfrlemibacc 6294	. . . . . . . . . 10 |- ( ph -> B C_ A )
10	9	unissd 3813	. . . . . . . . 9 |- ( ph -> U. B C_ U. A )
11	1	recsfval 6283	. . . . . . . . 9 |- recs ( F ) = U. A
12	10, 11	sseqtrrdi 3191	. . . . . . . 8 |- ( ph -> U. B C_ recs ( F ) )
13		dmss 4803	. . . . . . . 8 |- ( U. B C_ recs ( F ) -> dom U. B C_ dom recs ( F ) )
14	12, 13	syl 14	. . . . . . 7 |- ( ph -> dom U. B C_ dom recs ( F ) )
15	8, 14	eqsstrrd 3179	. . . . . 6 |- ( ph -> x C_ dom recs ( F ) )
16	15	sselda 3142	. . . . 5 |- ( ( ph /\ w e. x ) -> w e. dom recs ( F ) )
17	1	tfrlem9 6287	. . . . 5 |- ( w e. dom recs ( F ) -> (recs ( F ) ` w ) = ( F ` (recs ( F ) |` w ) ) )
18	16, 17	syl 14	. . . 4 |- ( ( ph /\ w e. x ) -> (recs ( F ) ` w ) = ( F ` (recs ( F ) |` w ) ) )
19	1	tfrlem7 6285	. . . . . 6 |- Fun recs ( F )
20	19	a1i 9	. . . . 5 |- ( ( ph /\ w e. x ) -> Fun recs ( F ) )
21	12	adantr 274	. . . . 5 |- ( ( ph /\ w e. x ) -> U. B C_ recs ( F ) )
22	8	eleq2d 2236	. . . . . 6 |- ( ph -> ( w e. dom U. B <-> w e. x ) )
23	22	biimpar 295	. . . . 5 |- ( ( ph /\ w e. x ) -> w e. dom U. B )
24		funssfv 5512	. . . . 5 |- ( ( Fun recs ( F ) /\ U. B C_ recs ( F ) /\ w e. dom U. B ) -> (recs ( F ) ` w ) = ( U. B ` w ) )
25	20, 21, 23, 24	syl3anc 1228	. . . 4 |- ( ( ph /\ w e. x ) -> (recs ( F ) ` w ) = ( U. B ` w ) )
26		eloni 4353	. . . . . . . . 9 |- ( x e. On -> Ord x )
27	4, 26	syl 14	. . . . . . . 8 |- ( ph -> Ord x )
28		ordelss 4357	. . . . . . . 8 |- ( ( Ord x /\ w e. x ) -> w C_ x )
29	27, 28	sylan 281	. . . . . . 7 |- ( ( ph /\ w e. x ) -> w C_ x )
30	8	adantr 274	. . . . . . 7 |- ( ( ph /\ w e. x ) -> dom U. B = x )
31	29, 30	sseqtrrd 3181	. . . . . 6 |- ( ( ph /\ w e. x ) -> w C_ dom U. B )
32		fun2ssres 5231	. . . . . 6 |- ( ( Fun recs ( F ) /\ U. B C_ recs ( F ) /\ w C_ dom U. B ) -> (recs ( F ) |` w ) = ( U. B |` w ) )
33	20, 21, 31, 32	syl3anc 1228	. . . . 5 |- ( ( ph /\ w e. x ) -> (recs ( F ) |` w ) = ( U. B |` w ) )
34	33	fveq2d 5490	. . . 4 |- ( ( ph /\ w e. x ) -> ( F ` (recs ( F ) |` w ) ) = ( F ` ( U. B |` w ) ) )
35	18, 25, 34	3eqtr3d 2206	. . 3 |- ( ( ph /\ w e. x ) -> ( U. B ` w ) = ( F ` ( U. B |` w ) ) )
36	35	ralrimiva 2539	. 2 |- ( ph -> A. w e. x ( U. B ` w ) = ( F ` ( U. B |` w ) ) )
37		fveq2 5486	. . . 4 |- ( u = w -> ( U. B ` u ) = ( U. B ` w ) )
38		reseq2 4879	. . . . 5 |- ( u = w -> ( U. B |` u ) = ( U. B |` w ) )
39	38	fveq2d 5490	. . . 4 |- ( u = w -> ( F ` ( U. B |` u ) ) = ( F ` ( U. B |` w ) ) )
40	37, 39	eqeq12d 2180	. . 3 |- ( u = w -> ( ( U. B ` u ) = ( F ` ( U. B |` u ) ) <-> ( U. B ` w ) = ( F ` ( U. B |` w ) ) ) )
41	40	cbvralv 2692	. 2 |- ( A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) <-> A. w e. x ( U. B ` w ) = ( F ` ( U. B |` w ) ) )
42	36, 41	sylibr 133	1 |- ( ph -> A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   A.wal 1341   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   _Vcvv 2726   u. cun 3114   C_ wss 3116   {csn 3576   <.cop 3579   U.cuni 3789   Ord word 4340   Oncon0 4341   dom cdm 4604   |` cres 4606   Fun wfun 5182   Fn wfn 5183   ` cfv 5188  recscrecs 6272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-recs 6273

This theorem is referenced by:  tfrlemiex  6299