Theorem tfrlemiubacc 5944

Description: The union of B satisfies the recursion rule (lemma for tfrlemi1 5946). (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 5942	. . . . . . . 8 |- ( ph -> U. B Fn x )
7		fndm 4998	. . . . . . . 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 5940	. . . . . . . . . 10 |- ( ph -> B C_ A )
10	9	unissd 3604	. . . . . . . . 9 |- ( ph -> U. B C_ U. A )
11	1	recsfval 5931	. . . . . . . . 9 |- recs ( F ) = U. A
12	10, 11	syl6sseqr 2992	. . . . . . . 8 |- ( ph -> U. B C_ recs ( F ) )
13		dmss 4534	. . . . . . . 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	eqsstr3d 2980	. . . . . 6 |- ( ph -> x C_ dom recs ( F ) )
16	15	sselda 2945	. . . . 5 |- ( ( ph /\ w e. x ) -> w e. dom recs ( F ) )
17	1	tfrlem9 5935	. . . . 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 5933	. . . . . 6 |- Fun recs ( F )
20	19	a1i 9	. . . . 5 |- ( ( ph /\ w e. x ) -> Fun recs ( F ) )
21	12	adantr 261	. . . . 5 |- ( ( ph /\ w e. x ) -> U. B C_ recs ( F ) )
22	8	eleq2d 2107	. . . . . 6 |- ( ph -> ( w e. dom U. B <-> w e. x ) )
23	22	biimpar 281	. . . . 5 |- ( ( ph /\ w e. x ) -> w e. dom U. B )
24		funssfv 5199	. . . . 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 1135	. . . 4 |- ( ( ph /\ w e. x ) -> (recs ( F ) ` w ) = ( U. B ` w ) )
26		eloni 4112	. . . . . . . . 9 |- ( x e. On -> Ord x )
27	4, 26	syl 14	. . . . . . . 8 |- ( ph -> Ord x )
28		ordelss 4116	. . . . . . . 8 |- ( ( Ord x /\ w e. x ) -> w C_ x )
29	27, 28	sylan 267	. . . . . . 7 |- ( ( ph /\ w e. x ) -> w C_ x )
30	8	adantr 261	. . . . . . 7 |- ( ( ph /\ w e. x ) -> dom U. B = x )
31	29, 30	sseqtr4d 2982	. . . . . 6 |- ( ( ph /\ w e. x ) -> w C_ dom U. B )
32		fun2ssres 4943	. . . . . 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 1135	. . . . 5 |- ( ( ph /\ w e. x ) -> (recs ( F ) |` w ) = ( U. B |` w ) )
34	33	fveq2d 5182	. . . 4 |- ( ( ph /\ w e. x ) -> ( F ` (recs ( F ) |` w ) ) = ( F ` ( U. B |` w ) ) )
35	18, 25, 34	3eqtr3d 2080	. . 3 |- ( ( ph /\ w e. x ) -> ( U. B ` w ) = ( F ` ( U. B |` w ) ) )
36	35	ralrimiva 2392	. 2 |- ( ph -> A. w e. x ( U. B ` w ) = ( F ` ( U. B |` w ) ) )
37		fveq2 5178	. . . 4 |- ( u = w -> ( U. B ` u ) = ( U. B ` w ) )
38		reseq2 4607	. . . . 5 |- ( u = w -> ( U. B |` u ) = ( U. B |` w ) )
39	38	fveq2d 5182	. . . 4 |- ( u = w -> ( F ` ( U. B |` u ) ) = ( F ` ( U. B |` w ) ) )
40	37, 39	eqeq12d 2054	. . 3 |- ( u = w -> ( ( U. B ` u ) = ( F ` ( U. B |` u ) ) <-> ( U. B ` w ) = ( F ` ( U. B |` w ) ) ) )
41	40	cbvralv 2533	. 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 137	1 |- ( ph -> A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   _Vcvv 2557   u. cun 2915   C_ wss 2917   {csn 3375   <.cop 3378   U.cuni 3580   Ord word 4099   Oncon0 4100   dom cdm 4345   |` cres 4347   Fun wfun 4896   Fn wfn 4897   ` cfv 4902  recscrecs 5919

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-recs 5920

This theorem is referenced by:  tfrlemiex  5945