Theorem tfrlemiubacc 6397

Description: The union of B satisfies the recursion rule (lemma for tfrlemi1 6399). (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 6395	. . . . . . . 8 |- ( ph -> U. B Fn x )
7		fndm 5358	. . . . . . . 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 6393	. . . . . . . . . 10 |- ( ph -> B C_ A )
10	9	unissd 3864	. . . . . . . . 9 |- ( ph -> U. B C_ U. A )
11	1	recsfval 6382	. . . . . . . . 9 |- recs ( F ) = U. A
12	10, 11	sseqtrrdi 3233	. . . . . . . 8 |- ( ph -> U. B C_ recs ( F ) )
13		dmss 4866	. . . . . . . 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 3221	. . . . . 6 |- ( ph -> x C_ dom recs ( F ) )
16	15	sselda 3184	. . . . 5 |- ( ( ph /\ w e. x ) -> w e. dom recs ( F ) )
17	1	tfrlem9 6386	. . . . 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 6384	. . . . . 6 |- Fun recs ( F )
20	19	a1i 9	. . . . 5 |- ( ( ph /\ w e. x ) -> Fun recs ( F ) )
21	12	adantr 276	. . . . 5 |- ( ( ph /\ w e. x ) -> U. B C_ recs ( F ) )
22	8	eleq2d 2266	. . . . . 6 |- ( ph -> ( w e. dom U. B <-> w e. x ) )
23	22	biimpar 297	. . . . 5 |- ( ( ph /\ w e. x ) -> w e. dom U. B )
24		funssfv 5587	. . . . 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 1249	. . . 4 |- ( ( ph /\ w e. x ) -> (recs ( F ) ` w ) = ( U. B ` w ) )
26		eloni 4411	. . . . . . . . 9 |- ( x e. On -> Ord x )
27	4, 26	syl 14	. . . . . . . 8 |- ( ph -> Ord x )
28		ordelss 4415	. . . . . . . 8 |- ( ( Ord x /\ w e. x ) -> w C_ x )
29	27, 28	sylan 283	. . . . . . 7 |- ( ( ph /\ w e. x ) -> w C_ x )
30	8	adantr 276	. . . . . . 7 |- ( ( ph /\ w e. x ) -> dom U. B = x )
31	29, 30	sseqtrrd 3223	. . . . . 6 |- ( ( ph /\ w e. x ) -> w C_ dom U. B )
32		fun2ssres 5302	. . . . . 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 1249	. . . . 5 |- ( ( ph /\ w e. x ) -> (recs ( F ) |` w ) = ( U. B |` w ) )
34	33	fveq2d 5565	. . . 4 |- ( ( ph /\ w e. x ) -> ( F ` (recs ( F ) |` w ) ) = ( F ` ( U. B |` w ) ) )
35	18, 25, 34	3eqtr3d 2237	. . 3 |- ( ( ph /\ w e. x ) -> ( U. B ` w ) = ( F ` ( U. B |` w ) ) )
36	35	ralrimiva 2570	. 2 |- ( ph -> A. w e. x ( U. B ` w ) = ( F ` ( U. B |` w ) ) )
37		fveq2 5561	. . . 4 |- ( u = w -> ( U. B ` u ) = ( U. B ` w ) )
38		reseq2 4942	. . . . 5 |- ( u = w -> ( U. B |` u ) = ( U. B |` w ) )
39	38	fveq2d 5565	. . . 4 |- ( u = w -> ( F ` ( U. B |` u ) ) = ( F ` ( U. B |` w ) ) )
40	37, 39	eqeq12d 2211	. . 3 |- ( u = w -> ( ( U. B ` u ) = ( F ` ( U. B |` u ) ) <-> ( U. B ` w ) = ( F ` ( U. B |` w ) ) ) )
41	40	cbvralv 2729	. 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 134	1 |- ( ph -> A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   A.wal 1362   = 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 3623   <.cop 3626   U.cuni 3840   Ord word 4398   Oncon0 4399   dom cdm 4664   |` cres 4666   Fun wfun 5253   Fn wfn 5254   ` cfv 5259  recscrecs 6371

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 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-recs 6372

This theorem is referenced by:  tfrlemiex  6398