Theorem tfrlem9 6528

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 4933	. . 3 |- ( B e. dom recs ( F ) -> ( B e. dom recs ( F ) <-> E. z <. B , z >. e. recs ( F ) ) )
2	1	ibi 176	. 2 |- ( B e. dom recs ( F ) -> E. z <. B , z >. e. recs ( F ) )
3		df-recs 6514	. . . . . 6 |- recs ( F ) = U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
4	3	eleq2i 2298	. . . . 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 3910	. . . . 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 184	. . . 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 5442	. . . . . . . . . . . . . 14 |- ( ( f Fn x /\ <. B , z >. e. f ) -> B e. x )
8		rspe 2582	. . . . . . . . . . . . . . . 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 2342	. . . . . . . . . . . . . . . . 17 |- ( f e. A <-> E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) )
11		elssuni 3926	. . . . . . . . . . . . . . . . . 18 |- ( f e. A -> f C_ U. A )
12	9	recsfval 6524	. . . . . . . . . . . . . . . . . 18 |- recs ( F ) = U. A
13	11, 12	sseqtrrdi 3277	. . . . . . . . . . . . . . . . 17 |- ( f e. A -> f C_ recs ( F ) )
14	10, 13	sylbir 135	. . . . . . . . . . . . . . . 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 5648	. . . . . . . . . . . . . . . . . . . 20 |- ( y = B -> ( f ` y ) = ( f ` B ) )
17		reseq2 5014	. . . . . . . . . . . . . . . . . . . . 21 |- ( y = B -> ( f |` y ) = ( f |` B ) )
18	17	fveq2d 5652	. . . . . . . . . . . . . . . . . . . 20 |- ( y = B -> ( F ` ( f |` y ) ) = ( F ` ( f |` B ) ) )
19	16, 18	eqeq12d 2246	. . . . . . . . . . . . . . . . . . 19 |- ( y = B -> ( ( f ` y ) = ( F ` ( f |` y ) ) <-> ( f ` B ) = ( F ` ( f |` B ) ) ) )
20	19	rspcv 2907	. . . . . . . . . . . . . . . . . 18 |- ( B e. x -> ( A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) -> ( f ` B ) = ( F ` ( f |` B ) ) ) )
21		fndm 5436	. . . . . . . . . . . . . . . . . . . . 21 |- ( f Fn x -> dom f = x )
22	21	eleq2d 2301	. . . . . . . . . . . . . . . . . . . 20 |- ( f Fn x -> ( B e. dom f <-> B e. x ) )
23	9	tfrlem7 6526	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- Fun recs ( F )
24		funssfv 5674	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( Fun recs ( F ) /\ f C_ recs ( F ) /\ B e. dom f ) -> (recs ( F ) ` B ) = ( f ` B ) )
25	23, 24	mp3an1 1361	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( f C_ recs ( F ) /\ B e. dom f ) -> (recs ( F ) ` B ) = ( f ` B ) )
26	25	adantrl 478	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( f C_ recs ( F ) /\ ( ( f Fn x /\ x e. On ) /\ B e. dom f ) ) -> (recs ( F ) ` B ) = ( f ` B ) )
27	21	eleq1d 2300	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( f Fn x -> ( dom f e. On <-> x e. On ) )
28		onelss 4490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( dom f e. On -> ( B e. dom f -> B C_ dom f ) )
29	27, 28	biimtrrdi 164	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( f Fn x -> ( x e. On -> ( B e. dom f -> B C_ dom f ) ) )
30	29	imp31 256	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( f Fn x /\ x e. On ) /\ B e. dom f ) -> B C_ dom f )
31		fun2ssres 5377	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( Fun recs ( F ) /\ f C_ recs ( F ) /\ B C_ dom f ) -> (recs ( F ) |` B ) = ( f |` B ) )
32	31	fveq2d 5652	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( Fun recs ( F ) /\ f C_ recs ( F ) /\ B C_ dom f ) -> ( F ` (recs ( F ) |` B ) ) = ( F ` ( f |` B ) ) )
33	23, 32	mp3an1 1361	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( f C_ recs ( F ) /\ B C_ dom f ) -> ( F ` (recs ( F ) |` B ) ) = ( F ` ( f |` B ) ) )
34	30, 33	sylan2 286	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2246	. . . . . . . . . . . . . . . . . . . . . . . . 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 382	. . . . . . . . . . . . . . . . . . . . . . . 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 364	. . . . . . . . . . . . . . . . . . . . . 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 170	. . . . . . . . . . . . . . . . . . 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 352	. . . . . . . . . . . . . . 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 369	. . . . . . . . . . . 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 115	. . . . . . . . . . 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 349	. . . . . . 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 2650	. . . . . 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 124	. . . . 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 1647	. . . 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 121	. . 3 |- ( <. B , z >. e. recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) )
58	57	exlimiv 1647	. 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 104   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2202   {cab 2217   A.wral 2511   E.wrex 2512   C_ wss 3201   <.cop 3676   U.cuni 3898   Oncon0 4466   dom cdm 4731   |` cres 4733   Fun wfun 5327   Fn wfn 5328   ` cfv 5333  recscrecs 6513

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-recs 6514

This theorem is referenced by:  tfr2a  6530  tfrlemiubacc  6539  tfr1onlemubacc  6555  tfrcllemubacc  6568