Theorem tfrlem9 5935

Description: Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)

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
tfrlem9	|- ( B e. dom recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) )

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

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

Proof of Theorem tfrlem9

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldm2g 4531	. . 3 |- ( B e. dom recs ( F ) -> ( B e. dom recs ( F ) <-> E. z <. B , z >. e. recs ( F ) ) )
2	1	ibi 165	. 2 |- ( B e. dom recs ( F ) -> E. z <. B , z >. e. recs ( F ) )
3		df-recs 5920	. . . . . 6 |- recs ( F ) = U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
4	3	eleq2i 2104	. . . . 5 |- ( <. B , z >. e. recs ( F ) <-> <. B , z >. e. U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } )
5		eluniab 3592	. . . . 5 |- ( <. B , z >. e. U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } <-> E. f ( <. B , z >. e. f /\ E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) ) )
6	4, 5	bitri 173	. . . 4 |- ( <. B , z >. e. recs ( F ) <-> E. f ( <. B , z >. e. f /\ E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) ) )
7		fnop 5002	. . . . . . . . . . . . . 14 |- ( ( f Fn x /\ <. B , z >. e. f ) -> B e. x )
8		rspe 2370	. . . . . . . . . . . . . . . 16 |- ( ( x e. On /\ ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) ) -> E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) )
9		tfrlem.1	. . . . . . . . . . . . . . . . . 18 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
10	9	abeq2i 2148	. . . . . . . . . . . . . . . . 17 |- ( f e. A <-> E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) )
11		elssuni 3608	. . . . . . . . . . . . . . . . . 18 |- ( f e. A -> f C_ U. A )
12	9	recsfval 5931	. . . . . . . . . . . . . . . . . 18 |- recs ( F ) = U. A
13	11, 12	syl6sseqr 2992	. . . . . . . . . . . . . . . . 17 |- ( f e. A -> f C_ recs ( F ) )
14	10, 13	sylbir 125	. . . . . . . . . . . . . . . 16 |- ( E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) -> f C_ recs ( F ) )
15	8, 14	syl 14	. . . . . . . . . . . . . . 15 |- ( ( x e. On /\ ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) ) -> f C_ recs ( F ) )
16		fveq2 5178	. . . . . . . . . . . . . . . . . . . 20 |- ( y = B -> ( f ` y ) = ( f ` B ) )
17		reseq2 4607	. . . . . . . . . . . . . . . . . . . . 21 |- ( y = B -> ( f |` y ) = ( f |` B ) )
18	17	fveq2d 5182	. . . . . . . . . . . . . . . . . . . 20 |- ( y = B -> ( F ` ( f |` y ) ) = ( F ` ( f |` B ) ) )
19	16, 18	eqeq12d 2054	. . . . . . . . . . . . . . . . . . 19 |- ( y = B -> ( ( f ` y ) = ( F ` ( f |` y ) ) <-> ( f ` B ) = ( F ` ( f |` B ) ) ) )
20	19	rspcv 2652	. . . . . . . . . . . . . . . . . 18 |- ( B e. x -> ( A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) -> ( f ` B ) = ( F ` ( f |` B ) ) ) )
21		fndm 4998	. . . . . . . . . . . . . . . . . . . . 21 |- ( f Fn x -> dom f = x )
22	21	eleq2d 2107	. . . . . . . . . . . . . . . . . . . 20 |- ( f Fn x -> ( B e. dom f <-> B e. x ) )
23	9	tfrlem7 5933	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- Fun recs ( F )
24		funssfv 5199	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( Fun recs ( F ) /\ f C_ recs ( F ) /\ B e. dom f ) -> (recs ( F ) ` B ) = ( f ` B ) )
25	23, 24	mp3an1 1219	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( f C_ recs ( F ) /\ B e. dom f ) -> (recs ( F ) ` B ) = ( f ` B ) )
26	25	adantrl 447	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( f C_ recs ( F ) /\ ( ( f Fn x /\ x e. On ) /\ B e. dom f ) ) -> (recs ( F ) ` B ) = ( f ` B ) )
27	21	eleq1d 2106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( f Fn x -> ( dom f e. On <-> x e. On ) )
28		onelss 4124	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( dom f e. On -> ( B e. dom f -> B C_ dom f ) )
29	27, 28	syl6bir 153	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( f Fn x -> ( x e. On -> ( B e. dom f -> B C_ dom f ) ) )
30	29	imp31 243	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( f Fn x /\ x e. On ) /\ B e. dom f ) -> B C_ dom f )
31		fun2ssres 4943	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( Fun recs ( F ) /\ f C_ recs ( F ) /\ B C_ dom f ) -> (recs ( F ) |` B ) = ( f |` B ) )
32	31	fveq2d 5182	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( Fun recs ( F ) /\ f C_ recs ( F ) /\ B C_ dom f ) -> ( F ` (recs ( F ) |` B ) ) = ( F ` ( f |` B ) ) )
33	23, 32	mp3an1 1219	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( f C_ recs ( F ) /\ B C_ dom f ) -> ( F ` (recs ( F ) |` B ) ) = ( F ` ( f |` B ) ) )
34	30, 33	sylan2 270	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( f C_ recs ( F ) /\ ( ( f Fn x /\ x e. On ) /\ B e. dom f ) ) -> ( F ` (recs ( F ) |` B ) ) = ( F ` ( f |` B ) ) )
35	26, 34	eqeq12d 2054	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( f C_ recs ( F ) /\ ( ( f Fn x /\ x e. On ) /\ B e. dom f ) ) -> ( (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) <-> ( f ` B ) = ( F ` ( f |` B ) ) ) )
36	35	exbiri 364	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( f C_ recs ( F ) -> ( ( ( f Fn x /\ x e. On ) /\ B e. dom f ) -> ( ( f ` B ) = ( F ` ( f |` B ) ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) )
37	36	com3l 75	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( f Fn x /\ x e. On ) /\ B e. dom f ) -> ( ( f ` B ) = ( F ` ( f |` B ) ) -> ( f C_ recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) )
38	37	exp31 346	. . . . . . . . . . . . . . . . . . . . . 22 |- ( f Fn x -> ( x e. On -> ( B e. dom f -> ( ( f ` B ) = ( F ` ( f |` B ) ) -> ( f C_ recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) ) ) )
39	38	com34 77	. . . . . . . . . . . . . . . . . . . . 21 |- ( f Fn x -> ( x e. On -> ( ( f ` B ) = ( F ` ( f |` B ) ) -> ( B e. dom f -> ( f C_ recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) ) ) )
40	39	com24 81	. . . . . . . . . . . . . . . . . . . 20 |- ( f Fn x -> ( B e. dom f -> ( ( f ` B ) = ( F ` ( f |` B ) ) -> ( x e. On -> ( f C_ recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) ) ) )
41	22, 40	sylbird 159	. . . . . . . . . . . . . . . . . . 19 |- ( f Fn x -> ( B e. x -> ( ( f ` B ) = ( F ` ( f |` B ) ) -> ( x e. On -> ( f C_ recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) ) ) )
42	41	com3l 75	. . . . . . . . . . . . . . . . . 18 |- ( B e. x -> ( ( f ` B ) = ( F ` ( f |` B ) ) -> ( f Fn x -> ( x e. On -> ( f C_ recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) ) ) )
43	20, 42	syld 40	. . . . . . . . . . . . . . . . 17 |- ( B e. x -> ( A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) -> ( f Fn x -> ( x e. On -> ( f C_ recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) ) ) )
44	43	com24 81	. . . . . . . . . . . . . . . 16 |- ( B e. x -> ( x e. On -> ( f Fn x -> ( A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) -> ( f C_ recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) ) ) )
45	44	imp4d 334	. . . . . . . . . . . . . . 15 |- ( B e. x -> ( ( x e. On /\ ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) ) -> ( f C_ recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) )
46	15, 45	mpdi 38	. . . . . . . . . . . . . 14 |- ( B e. x -> ( ( x e. On /\ ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) )
47	7, 46	syl 14	. . . . . . . . . . . . 13 |- ( ( f Fn x /\ <. B , z >. e. f ) -> ( ( x e. On /\ ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) )
48	47	exp4d 351	. . . . . . . . . . . 12 |- ( ( f Fn x /\ <. B , z >. e. f ) -> ( x e. On -> ( f Fn x -> ( A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) ) )
49	48	ex 108	. . . . . . . . . . 11 |- ( f Fn x -> ( <. B , z >. e. f -> ( x e. On -> ( f Fn x -> ( A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) ) ) )
50	49	com4r 80	. . . . . . . . . 10 |- ( f Fn x -> ( f Fn x -> ( <. B , z >. e. f -> ( x e. On -> ( A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) ) ) )
51	50	pm2.43i 43	. . . . . . . . 9 |- ( f Fn x -> ( <. B , z >. e. f -> ( x e. On -> ( A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) ) )
52	51	com3l 75	. . . . . . . 8 |- ( <. B , z >. e. f -> ( x e. On -> ( f Fn x -> ( A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) ) )
53	52	imp4a 331	. . . . . . 7 |- ( <. B , z >. e. f -> ( x e. On -> ( ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) ) )
54	53	rexlimdv 2432	. . . . . 6 |- ( <. B , z >. e. f -> ( E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) ) )
55	54	imp 115	. . . . 5 |- ( ( <. B , z >. e. f /\ E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) )
56	55	exlimiv 1489	. . . 4 |- ( E. f ( <. B , z >. e. f /\ E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) )
57	6, 56	sylbi 114	. . 3 |- ( <. B , z >. e. recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) )
58	57	exlimiv 1489	. 2 |- ( E. z <. B , z >. e. recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) )
59	2, 58	syl 14	1 |- ( B e. dom recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   E.wex 1381   e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   C_ wss 2917   <.cop 3378   U.cuni 3580   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-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-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-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  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-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  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-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-recs 5920

This theorem is referenced by:  tfr2a  5936  tfrlemiubacc  5944