Theorem tfrlem9a 7482

Description: Lemma for transfinite recursion. Without using ax-rep 4771, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)

Hypothesis

Ref	Expression
tfrlem.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }

Assertion

Ref	Expression
tfrlem9a	|- ( B e. dom recs ( F ) -> (recs ( F ) |` B ) e. _V )

Distinct variable groups:   x, f, y, B   f, F, x, y

Allowed substitution hints:   A( x, y, f)

Proof of Theorem tfrlem9a

Dummy variables g z a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . . 5 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
2	1	tfrlem7 7479	. . . 4 |- Fun recs ( F )
3		funfvop 6329	. . . 4 |- ( ( Fun recs ( F ) /\ B e. dom recs ( F ) ) -> <. B , (recs ( F ) ` B ) >. e. recs ( F ) )
4	2, 3	mpan 706	. . 3 |- ( B e. dom recs ( F ) -> <. B , (recs ( F ) ` B ) >. e. recs ( F ) )
5	1	recsfval 7477	. . . . 5 |- recs ( F ) = U. A
6	5	eleq2i 2693	. . . 4 |- ( <. B , (recs ( F ) ` B ) >. e. recs ( F ) <-> <. B , (recs ( F ) ` B ) >. e. U. A )
7		eluni 4439	. . . 4 |- ( <. B , (recs ( F ) ` B ) >. e. U. A <-> E. g ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) )
8	6, 7	bitri 264	. . 3 |- ( <. B , (recs ( F ) ` B ) >. e. recs ( F ) <-> E. g ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) )
9	4, 8	sylib 208	. 2 |- ( B e. dom recs ( F ) -> E. g ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) )
10		simprr 796	. . . 4 |- ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) -> g e. A )
11		vex 3203	. . . . 5 |- g e. _V
12	1, 11	tfrlem3a 7473	. . . 4 |- ( g e. A <-> E. z e. On ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) )
13	10, 12	sylib 208	. . 3 |- ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) -> E. z e. On ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) )
14	2	a1i 11	. . . . . . . 8 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> Fun recs ( F ) )
15		simplrr 801	. . . . . . . . . 10 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> g e. A )
16		elssuni 4467	. . . . . . . . . 10 |- ( g e. A -> g C_ U. A )
17	15, 16	syl 17	. . . . . . . . 9 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> g C_ U. A )
18	17, 5	syl6sseqr 3652	. . . . . . . 8 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> g C_ recs ( F ) )
19		fndm 5990	. . . . . . . . . . . 12 |- ( g Fn z -> dom g = z )
20	19	ad2antll 765	. . . . . . . . . . 11 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> dom g = z )
21		simprl 794	. . . . . . . . . . 11 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> z e. On )
22	20, 21	eqeltrd 2701	. . . . . . . . . 10 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> dom g e. On )
23		eloni 5733	. . . . . . . . . 10 |- ( dom g e. On -> Ord dom g )
24	22, 23	syl 17	. . . . . . . . 9 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> Ord dom g )
25		simpll 790	. . . . . . . . . 10 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> B e. dom recs ( F ) )
26		fvexd 6203	. . . . . . . . . 10 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> (recs ( F ) ` B ) e. _V )
27		simplrl 800	. . . . . . . . . . 11 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> <. B , (recs ( F ) ` B ) >. e. g )
28		df-br 4654	. . . . . . . . . . 11 |- ( B g (recs ( F ) ` B ) <-> <. B , (recs ( F ) ` B ) >. e. g )
29	27, 28	sylibr 224	. . . . . . . . . 10 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> B g (recs ( F ) ` B ) )
30		breldmg 5330	. . . . . . . . . 10 |- ( ( B e. dom recs ( F ) /\ (recs ( F ) ` B ) e. _V /\ B g (recs ( F ) ` B ) ) -> B e. dom g )
31	25, 26, 29, 30	syl3anc 1326	. . . . . . . . 9 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> B e. dom g )
32		ordelss 5739	. . . . . . . . 9 |- ( ( Ord dom g /\ B e. dom g ) -> B C_ dom g )
33	24, 31, 32	syl2anc 693	. . . . . . . 8 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> B C_ dom g )
34		fun2ssres 5931	. . . . . . . 8 |- ( ( Fun recs ( F ) /\ g C_ recs ( F ) /\ B C_ dom g ) -> (recs ( F ) |` B ) = ( g |` B ) )
35	14, 18, 33, 34	syl3anc 1326	. . . . . . 7 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> (recs ( F ) |` B ) = ( g |` B ) )
36	11	resex 5443	. . . . . . . 8 |- ( g |` B ) e. _V
37	36	a1i 11	. . . . . . 7 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> ( g |` B ) e. _V )
38	35, 37	eqeltrd 2701	. . . . . 6 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ ( z e. On /\ g Fn z ) ) -> (recs ( F ) |` B ) e. _V )
39	38	expr 643	. . . . 5 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ z e. On ) -> ( g Fn z -> (recs ( F ) |` B ) e. _V ) )
40	39	adantrd 484	. . . 4 |- ( ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) /\ z e. On ) -> ( ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) -> (recs ( F ) |` B ) e. _V ) )
41	40	rexlimdva 3031	. . 3 |- ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) -> ( E. z e. On ( g Fn z /\ A. a e. z ( g ` a ) = ( F ` ( g |` a ) ) ) -> (recs ( F ) |` B ) e. _V ) )
42	13, 41	mpd 15	. 2 |- ( ( B e. dom recs ( F ) /\ ( <. B , (recs ( F ) ` B ) >. e. g /\ g e. A ) ) -> (recs ( F ) |` B ) e. _V )
43	9, 42	exlimddv 1863	1 |- ( B e. dom recs ( F ) -> (recs ( F ) |` B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   <.cop 4183   U.cuni 4436   class class class wbr 4653   dom cdm 5114   |` cres 5116   Ord word 5722   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  recscrecs 7467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  tfrlem15  7488  tfrlem16  7489  rdgseg  7518