Theorem rdgivallem 5908

Description: Value of the recursive definition generator. Lemma for rdgival 5909 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)

Assertion

Ref	Expression
rdgivallem	|- (( F Fn _V /\ A e. V /\ B e. On) -> ( rec( F , A) ` B) = (A u. U_x e. B ( F ` (( rec( F , A) |` B) ` x))))

Distinct variable groups:   x,A   x,B   x, F   x, V

Proof of Theorem rdgivallem

Dummy variables g y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-irdg 5897	. . . 4 |- rec( F , A) = recs((g e. _V |-> (A u. U_x e. dom g( F ` (g ` x)))))
2		rdgruledefgg 5902	. . . . 5 |- (( F Fn _V /\ A e. V) -> ( Fun (g e. _V |-> (A u. U_x e. dom g( F ` (g ` x)))) /\ ((g e. _V |-> (A u. U_x e. dom g( F ` (g ` x)))) ` y) e. _V))
3	2	alrimiv 1751	. . . 4 |- (( F Fn _V /\ A e. V) -> A.y( Fun (g e. _V |-> (A u. U_x e. dom g( F ` (g ` x)))) /\ ((g e. _V |-> (A u. U_x e. dom g( F ` (g ` x)))) ` y) e. _V))
4	1, 3	tfri2d 5891	. . 3 |- ((( F Fn _V /\ A e. V) /\ B e. On) -> ( rec( F , A) ` B) = ((g e. _V |-> (A u. U_x e. dom g( F ` (g ` x)))) ` ( rec( F , A) |` B)))
5	4	3impa 1098	. 2 |- (( F Fn _V /\ A e. V /\ B e. On) -> ( rec( F , A) ` B) = ((g e. _V |-> (A u. U_x e. dom g( F ` (g ` x)))) ` ( rec( F , A) |` B)))
6		eqidd 2038	. . 3 |- (( F Fn _V /\ A e. V /\ B e. On) -> (g e. _V |-> (A u. U_x e. dom g( F ` (g ` x)))) = (g e. _V |-> (A u. U_x e. dom g( F ` (g ` x)))))
7		dmeq 4478	. . . . . 6 |- (g = ( rec( F , A) |` B) -> dom g = dom ( rec( F , A) |` B))
8		onss 4185	. . . . . . . . 9 |- (B e. On -> B C_ On)
9	8	3ad2ant3 926	. . . . . . . 8 |- (( F Fn _V /\ A e. V /\ B e. On) -> B C_ On)
10		rdgifnon 5906	. . . . . . . . . 10 |- (( F Fn _V /\ A e. V) -> rec( F , A) Fn On)
11		fndm 4941	. . . . . . . . . 10 |- ( rec( F , A) Fn On -> dom rec( F , A) = On)
12	10, 11	syl 14	. . . . . . . . 9 |- (( F Fn _V /\ A e. V) -> dom rec( F , A) = On)
13	12	3adant3 923	. . . . . . . 8 |- (( F Fn _V /\ A e. V /\ B e. On) -> dom rec( F , A) = On)
14	9, 13	sseqtr4d 2976	. . . . . . 7 |- (( F Fn _V /\ A e. V /\ B e. On) -> B C_ dom rec( F , A))
15		ssdmres 4576	. . . . . . 7 |- (B C_ dom rec( F , A) <-> dom ( rec( F , A) |` B) = B)
16	14, 15	sylib 127	. . . . . 6 |- (( F Fn _V /\ A e. V /\ B e. On) -> dom ( rec( F , A) |` B) = B)
17	7, 16	sylan9eqr 2091	. . . . 5 |- ((( F Fn _V /\ A e. V /\ B e. On) /\ g = ( rec( F , A) |` B)) -> dom g = B)
18		fveq1 5120	. . . . . . 7 |- (g = ( rec( F , A) |` B) -> (g ` x) = (( rec( F , A) |` B) ` x))
19	18	fveq2d 5125	. . . . . 6 |- (g = ( rec( F , A) |` B) -> ( F ` (g ` x)) = ( F ` (( rec( F , A) |` B) ` x)))
20	19	adantl 262	. . . . 5 |- ((( F Fn _V /\ A e. V /\ B e. On) /\ g = ( rec( F , A) |` B)) -> ( F ` (g ` x)) = ( F ` (( rec( F , A) |` B) ` x)))
21	17, 20	iuneq12d 3672	. . . 4 |- ((( F Fn _V /\ A e. V /\ B e. On) /\ g = ( rec( F , A) |` B)) -> U_x e. dom g( F ` (g ` x)) = U_x e. B ( F ` (( rec( F , A) |` B) ` x)))
22	21	uneq2d 3091	. . 3 |- ((( F Fn _V /\ A e. V /\ B e. On) /\ g = ( rec( F , A) |` B)) -> (A u. U_x e. dom g( F ` (g ` x))) = (A u. U_x e. B ( F ` (( rec( F , A) |` B) ` x))))
23		rdgfun 5900	. . . . 5 |- Fun rec( F , A)
24		resfunexg 5325	. . . . 5 |- (( Fun rec( F , A) /\ B e. On) -> ( rec( F , A) |` B) e. _V)
25	23, 24	mpan 400	. . . 4 |- (B e. On -> ( rec( F , A) |` B) e. _V)
26	25	3ad2ant3 926	. . 3 |- (( F Fn _V /\ A e. V /\ B e. On) -> ( rec( F , A) |` B) e. _V)
27		simpr 103	. . . . . 6 |- (( F Fn _V /\ B e. On) -> B e. On)
28		vex 2554	. . . . . . . . . 10 |- x e. _V
29		fvexg 5137	. . . . . . . . . 10 |- ((( rec( F , A) |` B) e. _V /\ x e. _V) -> (( rec( F , A) |` B) ` x) e. _V)
30	25, 28, 29	sylancl 392	. . . . . . . . 9 |- (B e. On -> (( rec( F , A) |` B) ` x) e. _V)
31	30	ralrimivw 2387	. . . . . . . 8 |- (B e. On -> A.x e. B (( rec( F , A) |` B) ` x) e. _V)
32	31	adantl 262	. . . . . . 7 |- (( F Fn _V /\ B e. On) -> A.x e. B (( rec( F , A) |` B) ` x) e. _V)
33		funfvex 5135	. . . . . . . . . . 11 |- (( Fun F /\ (( rec( F , A) |` B) ` x) e. dom F) -> ( F ` (( rec( F , A) |` B) ` x)) e. _V)
34	33	funfni 4942	. . . . . . . . . 10 |- (( F Fn _V /\ (( rec( F , A) |` B) ` x) e. _V) -> ( F ` (( rec( F , A) |` B) ` x)) e. _V)
35	34	ex 108	. . . . . . . . 9 |- ( F Fn _V -> ((( rec( F , A) |` B) ` x) e. _V -> ( F ` (( rec( F , A) |` B) ` x)) e. _V))
36	35	ralimdv 2382	. . . . . . . 8 |- ( F Fn _V -> (A.x e. B (( rec( F , A) |` B) ` x) e. _V -> A.x e. B ( F ` (( rec( F , A) |` B) ` x)) e. _V))
37	36	adantr 261	. . . . . . 7 |- (( F Fn _V /\ B e. On) -> (A.x e. B (( rec( F , A) |` B) ` x) e. _V -> A.x e. B ( F ` (( rec( F , A) |` B) ` x)) e. _V))
38	32, 37	mpd 13	. . . . . 6 |- (( F Fn _V /\ B e. On) -> A.x e. B ( F ` (( rec( F , A) |` B) ` x)) e. _V)
39		iunexg 5688	. . . . . 6 |- ((B e. On /\ A.x e. B ( F ` (( rec( F , A) |` B) ` x)) e. _V) -> U_x e. B ( F ` (( rec( F , A) |` B) ` x)) e. _V)
40	27, 38, 39	syl2anc 391	. . . . 5 |- (( F Fn _V /\ B e. On) -> U_x e. B ( F ` (( rec( F , A) |` B) ` x)) e. _V)
41	40	3adant2 922	. . . 4 |- (( F Fn _V /\ A e. V /\ B e. On) -> U_x e. B ( F ` (( rec( F , A) |` B) ` x)) e. _V)
42		unexg 4144	. . . . . 6 |- ((A e. V /\ U_x e. B ( F ` (( rec( F , A) |` B) ` x)) e. _V) -> (A u. U_x e. B ( F ` (( rec( F , A) |` B) ` x))) e. _V)
43	42	ex 108	. . . . 5 |- (A e. V -> ( U_x e. B ( F ` (( rec( F , A) |` B) ` x)) e. _V -> (A u. U_x e. B ( F ` (( rec( F , A) |` B) ` x))) e. _V))
44	43	3ad2ant2 925	. . . 4 |- (( F Fn _V /\ A e. V /\ B e. On) -> ( U_x e. B ( F ` (( rec( F , A) |` B) ` x)) e. _V -> (A u. U_x e. B ( F ` (( rec( F , A) |` B) ` x))) e. _V))
45	41, 44	mpd 13	. . 3 |- (( F Fn _V /\ A e. V /\ B e. On) -> (A u. U_x e. B ( F ` (( rec( F , A) |` B) ` x))) e. _V)
46	6, 22, 26, 45	fvmptd 5196	. 2 |- (( F Fn _V /\ A e. V /\ B e. On) -> ((g e. _V |-> (A u. U_x e. dom g( F ` (g ` x)))) ` ( rec( F , A) |` B)) = (A u. U_x e. B ( F ` (( rec( F , A) |` B) ` x))))
47	5, 46	eqtrd 2069	1 |- (( F Fn _V /\ A e. V /\ B e. On) -> ( rec( F , A) ` B) = (A u. U_x e. B ( F ` (( rec( F , A) |` B) ` x))))

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 884   = wceq 1242   e. wcel 1390  A.wral 2300   _Vcvv 2551   u. cun 2909   C_ wss 2911   U_ciun 3648   |-> cmpt 3809   Oncon0 4066   dom cdm 4288   |` cres 4290   Fun wfun 4839   Fn wfn 4840   ` cfv 4845   reccrdg 5896

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-in1 544  ax-in2 545  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-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  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-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  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-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-recs 5861  df-irdg 5897

This theorem is referenced by:  rdgival  5909  rdgon  5913