Theorem tfrlem9 6209

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 4730	. . 3 |- ( B e. dom recs ( F ) -> ( B e. dom recs ( F ) <-> E. z <. B , z >. e. recs ( F ) ) )
2	1	ibi 175	. 2 |- ( B e. dom recs ( F ) -> E. z <. B , z >. e. recs ( F ) )
3		df-recs 6195	. . . . . 6 |- recs ( F ) = U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
4	3	eleq2i 2204	. . . . 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 3743	. . . . 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 183	. . . 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 5221	. . . . . . . . . . . . . 14 |- ( ( f Fn x /\ <. B , z >. e. f ) -> B e. x )
8		rspe 2479	. . . . . . . . . . . . . . . 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 2248	. . . . . . . . . . . . . . . . 17 |- ( f e. A <-> E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) )
11		elssuni 3759	. . . . . . . . . . . . . . . . . 18 |- ( f e. A -> f C_ U. A )
12	9	recsfval 6205	. . . . . . . . . . . . . . . . . 18 |- recs ( F ) = U. A
13	11, 12	sseqtrrdi 3141	. . . . . . . . . . . . . . . . 17 |- ( f e. A -> f C_ recs ( F ) )
14	10, 13	sylbir 134	. . . . . . . . . . . . . . . 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 5414	. . . . . . . . . . . . . . . . . . . 20 |- ( y = B -> ( f ` y ) = ( f ` B ) )
17		reseq2 4809	. . . . . . . . . . . . . . . . . . . . 21 |- ( y = B -> ( f |` y ) = ( f |` B ) )
18	17	fveq2d 5418	. . . . . . . . . . . . . . . . . . . 20 |- ( y = B -> ( F ` ( f |` y ) ) = ( F ` ( f |` B ) ) )
19	16, 18	eqeq12d 2152	. . . . . . . . . . . . . . . . . . 19 |- ( y = B -> ( ( f ` y ) = ( F ` ( f |` y ) ) <-> ( f ` B ) = ( F ` ( f |` B ) ) ) )
20	19	rspcv 2780	. . . . . . . . . . . . . . . . . 18 |- ( B e. x -> ( A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) -> ( f ` B ) = ( F ` ( f |` B ) ) ) )
21		fndm 5217	. . . . . . . . . . . . . . . . . . . . 21 |- ( f Fn x -> dom f = x )
22	21	eleq2d 2207	. . . . . . . . . . . . . . . . . . . 20 |- ( f Fn x -> ( B e. dom f <-> B e. x ) )
23	9	tfrlem7 6207	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- Fun recs ( F )
24		funssfv 5440	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( Fun recs ( F ) /\ f C_ recs ( F ) /\ B e. dom f ) -> (recs ( F ) ` B ) = ( f ` B ) )
25	23, 24	mp3an1 1302	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( f C_ recs ( F ) /\ B e. dom f ) -> (recs ( F ) ` B ) = ( f ` B ) )
26	25	adantrl 469	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( f C_ recs ( F ) /\ ( ( f Fn x /\ x e. On ) /\ B e. dom f ) ) -> (recs ( F ) ` B ) = ( f ` B ) )
27	21	eleq1d 2206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( f Fn x -> ( dom f e. On <-> x e. On ) )
28		onelss 4304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( dom f e. On -> ( B e. dom f -> B C_ dom f ) )
29	27, 28	syl6bir 163	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( f Fn x -> ( x e. On -> ( B e. dom f -> B C_ dom f ) ) )
30	29	imp31 254	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( f Fn x /\ x e. On ) /\ B e. dom f ) -> B C_ dom f )
31		fun2ssres 5161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( Fun recs ( F ) /\ f C_ recs ( F ) /\ B C_ dom f ) -> (recs ( F ) |` B ) = ( f |` B ) )
32	31	fveq2d 5418	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( Fun recs ( F ) /\ f C_ recs ( F ) /\ B C_ dom f ) -> ( F ` (recs ( F ) |` B ) ) = ( F ` ( f |` B ) ) )
33	23, 32	mp3an1 1302	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( f C_ recs ( F ) /\ B C_ dom f ) -> ( F ` (recs ( F ) |` B ) ) = ( F ` ( f |` B ) ) )
34	30, 33	sylan2 284	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2152	. . . . . . . . . . . . . . . . . . . . . . . . 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 379	. . . . . . . . . . . . . . . . . . . . . . . 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 81	. . . . . . . . . . . . . . . . . . . . . . 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 361	. . . . . . . . . . . . . . . . . . . . . 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 83	. . . . . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . . . . 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 169	. . . . . . . . . . . . . . . . . . 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 81	. . . . . . . . . . . . . . . . . 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 45	. . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . 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 349	. . . . . . . . . . . . . . 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 43	. . . . . . . . . . . . . 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 366	. . . . . . . . . . . 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 114	. . . . . . . . . . 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 86	. . . . . . . . . 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 49	. . . . . . . . 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 81	. . . . . . . 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 346	. . . . . . 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 2546	. . . . . 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 123	. . . . 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 1577	. . . 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 120	. . 3 |- ( <. B , z >. e. recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) )
58	57	exlimiv 1577	. 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 103   /\ w3a 962   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   C_ wss 3066   <.cop 3525   U.cuni 3731   Oncon0 4280   dom cdm 4534   |` cres 4536   Fun wfun 5112   Fn wfn 5113   ` cfv 5118  recscrecs 6194

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126  df-recs 6195

This theorem is referenced by:  tfr2a  6211  tfrlemiubacc  6220  tfr1onlemubacc  6236  tfrcllemubacc  6249