Theorem tfrlem9 5876

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 4474	. . 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 5861	. . . . . 6 |- recs( F) = U. {f | E.x e. On (f Fn x /\ A.y e. x (f ` y) = ( F ` (f |` y))) }
4	3	eleq2i 2101	. . . . 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 3583	. . . . 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 4945	. . . . . . . . . . . . . 14 |- ((f Fn x /\ <.B , z >. e. f) -> B e. x)
8		rspe 2364	. . . . . . . . . . . . . . . 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 2145	. . . . . . . . . . . . . . . . 17 |- (f e. A <-> E.x e. On (f Fn x /\ A.y e. x (f ` y) = ( F ` (f |` y))))
11		elssuni 3599	. . . . . . . . . . . . . . . . . 18 |- (f e. A -> f C_ U.A)
12	9	recsfval 5872	. . . . . . . . . . . . . . . . . 18 |- recs( F) = U.A
13	11, 12	syl6sseqr 2986	. . . . . . . . . . . . . . . . 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 5121	. . . . . . . . . . . . . . . . . . . 20 |- (y = B -> (f ` y) = (f ` B))
17		reseq2 4550	. . . . . . . . . . . . . . . . . . . . 21 |- (y = B -> (f |` y) = (f |` B))
18	17	fveq2d 5125	. . . . . . . . . . . . . . . . . . . 20 |- (y = B -> ( F ` (f |` y)) = ( F ` (f |` B)))
19	16, 18	eqeq12d 2051	. . . . . . . . . . . . . . . . . . 19 |- (y = B -> ((f ` y) = ( F ` (f |` y)) <-> (f ` B) = ( F ` (f |` B))))
20	19	rspcv 2646	. . . . . . . . . . . . . . . . . 18 |- (B e. x -> (A.y e. x (f ` y) = ( F ` (f |` y)) -> (f ` B) = ( F ` (f |` B))))
21		fndm 4941	. . . . . . . . . . . . . . . . . . . . 21 |- (f Fn x -> dom f = x)
22	21	eleq2d 2104	. . . . . . . . . . . . . . . . . . . 20 |- (f Fn x -> (B e. dom f <-> B e. x))
23	9	tfrlem7 5874	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- Fun recs( F)
24		funssfv 5142	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (( Fun recs( F) /\ f C_ recs( F) /\ B e. dom f) -> (recs( F) ` B) = (f ` B))
25	23, 24	mp3an1 1218	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (f Fn x -> ( dom f e. On <-> x e. On))
28		onelss 4090	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4886	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (( Fun recs( F) /\ f C_ recs( F) /\ B C_ dom f) -> (recs( F) |` B) = (f |` B))
32	31	fveq2d 5125	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (( Fun recs( F) /\ f C_ recs( F) /\ B C_ dom f) -> ( F ` (recs( F) |` B)) = ( F ` (f |` B)))
33	23, 32	mp3an1 1218	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2051	. . . . . . . . . . . . . . . . . . . . . . . . 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 2426	. . . . . 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 1486	. . . 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 1486	. 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 884   = wceq 1242  E.wex 1378   e. wcel 1390   {cab 2023  A.wral 2300  E.wrex 2301   C_ wss 2911   <.cop 3370   U.cuni 3571   Oncon0 4066   dom cdm 4288   |` cres 4290   Fun wfun 4839   Fn wfn 4840   ` cfv 4845  recscrecs 5860

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-recs 5861

This theorem is referenced by:  tfr2a  5877  tfrlemiubacc  5885