Theorem r1val1 8244

Description: The value of the cumulative hierarchy of sets function expressed recursively. Theorem 7Q of [Enderton] p. 202. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
r1val1	|- ( A e. dom R1 -> ( R1 ` A ) = U_ x e. A ~P ( R1 ` x ) )

Distinct variable group:   x, A

Proof of Theorem r1val1

Step	Hyp	Ref	Expression
1		simpr 467	. . . . . 6 |- ( ( A e. dom R1 /\ A = (/) ) -> A = (/) )
2	1	fveq2d 5852	. . . . 5 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) = ( R1 ` (/) ) )
3		r10 8226	. . . . 5 |- ( R1 ` (/) ) = (/)
4	2, 3	syl6eq 2502	. . . 4 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) = (/) )
5		0ss 3731	. . . . 5 |- (/) C_ U_ x e. A ~P ( R1 ` x )
6	5	a1i 11	. . . 4 |- ( ( A e. dom R1 /\ A = (/) ) -> (/) C_ U_ x e. A ~P ( R1 ` x ) )
7	4, 6	eqsstrd 3434	. . 3 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
8		nfv 1765	. . . . 5 |- F/ x A e. dom R1
9		nfcv 2593	. . . . . 6 |- F/_ x ( R1 ` A )
10		nfiu1 4278	. . . . . 6 |- F/_ x U_ x e. A ~P ( R1 ` x )
11	9, 10	nfss 3393	. . . . 5 |- F/ x ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x )
12		simpr 467	. . . . . . . . . 10 |- ( ( A e. dom R1 /\ A = suc x ) -> A = suc x )
13	12	fveq2d 5852	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) = ( R1 ` suc x ) )
14		eleq1 2518	. . . . . . . . . . . 12 |- ( A = suc x -> ( A e. dom R1 <-> suc x e. dom R1 ) )
15	14	biimpac 493	. . . . . . . . . . 11 |- ( ( A e. dom R1 /\ A = suc x ) -> suc x e. dom R1 )
16		r1funlim 8224	. . . . . . . . . . . . 13 |- ( Fun R1 /\ Lim dom R1 )
17	16	simpri 468	. . . . . . . . . . . 12 |- Lim dom R1
18		limsuc 6664	. . . . . . . . . . . 12 |- ( Lim dom R1 -> ( x e. dom R1 <-> suc x e. dom R1 ) )
19	17, 18	ax-mp 5	. . . . . . . . . . 11 |- ( x e. dom R1 <-> suc x e. dom R1 )
20	15, 19	sylibr 217	. . . . . . . . . 10 |- ( ( A e. dom R1 /\ A = suc x ) -> x e. dom R1 )
21		r1sucg 8227	. . . . . . . . . 10 |- ( x e. dom R1 -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
22	20, 21	syl 17	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
23	13, 22	eqtrd 2486	. . . . . . . 8 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) = ~P ( R1 ` x ) )
24		vex 3016	. . . . . . . . . . 11 |- x e. _V
25	24	sucid 5481	. . . . . . . . . 10 |- x e. suc x
26	25, 12	syl5eleqr 2537	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> x e. A )
27		ssiun2 4291	. . . . . . . . 9 |- ( x e. A -> ~P ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
28	26, 27	syl 17	. . . . . . . 8 |- ( ( A e. dom R1 /\ A = suc x ) -> ~P ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
29	23, 28	eqsstrd 3434	. . . . . . 7 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
30	29	ex 440	. . . . . 6 |- ( A e. dom R1 -> ( A = suc x -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) ) )
31	30	a1d 26	. . . . 5 |- ( A e. dom R1 -> ( x e. On -> ( A = suc x -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) ) ) )
32	8, 11, 31	rexlimd 2847	. . . 4 |- ( A e. dom R1 -> ( E. x e. On A = suc x -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) ) )
33	32	imp 435	. . 3 |- ( ( A e. dom R1 /\ E. x e. On A = suc x ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
34		r1limg 8229	. . . . 5 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` A ) = U_ x e. A ( R1 ` x ) )
35		r1tr 8234	. . . . . . . . 9 |- Tr ( R1 ` x )
36		dftr4 4474	. . . . . . . . 9 |- ( Tr ( R1 ` x ) <-> ( R1 ` x ) C_ ~P ( R1 ` x ) )
37	35, 36	mpbi 213	. . . . . . . 8 |- ( R1 ` x ) C_ ~P ( R1 ` x )
38	37	a1i 11	. . . . . . 7 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` x ) C_ ~P ( R1 ` x ) )
39	38	ralrimivw 2791	. . . . . 6 |- ( ( A e. dom R1 /\ Lim A ) -> A. x e. A ( R1 ` x ) C_ ~P ( R1 ` x ) )
40		ss2iun 4264	. . . . . 6 |- ( A. x e. A ( R1 ` x ) C_ ~P ( R1 ` x ) -> U_ x e. A ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
41	39, 40	syl 17	. . . . 5 |- ( ( A e. dom R1 /\ Lim A ) -> U_ x e. A ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
42	34, 41	eqsstrd 3434	. . . 4 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
43	42	adantrl 727	. . 3 |- ( ( A e. dom R1 /\ ( A e. _V /\ Lim A ) ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
44		limord 5461	. . . . . . 7 |- ( Lim dom R1 -> Ord dom R1 )
45	17, 44	ax-mp 5	. . . . . 6 |- Ord dom R1
46		ordsson 6604	. . . . . 6 |- ( Ord dom R1 -> dom R1 C_ On )
47	45, 46	ax-mp 5	. . . . 5 |- dom R1 C_ On
48	47	sseli 3396	. . . 4 |- ( A e. dom R1 -> A e. On )
49		onzsl 6661	. . . 4 |- ( A e. On <-> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
50	48, 49	sylib 201	. . 3 |- ( A e. dom R1 -> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
51	7, 33, 43, 50	mpjao3dan 1339	. 2 |- ( A e. dom R1 -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
52		ordtr1 5445	. . . . . . . 8 |- ( Ord dom R1 -> ( ( x e. A /\ A e. dom R1 ) -> x e. dom R1 ) )
53	45, 52	ax-mp 5	. . . . . . 7 |- ( ( x e. A /\ A e. dom R1 ) -> x e. dom R1 )
54	53	ancoms 459	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> x e. dom R1 )
55	54, 21	syl 17	. . . . 5 |- ( ( A e. dom R1 /\ x e. A ) -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
56		simpr 467	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> x e. A )
57		ordelord 5424	. . . . . . . . . 10 |- ( ( Ord dom R1 /\ A e. dom R1 ) -> Ord A )
58	45, 57	mpan 681	. . . . . . . . 9 |- ( A e. dom R1 -> Ord A )
59	58	adantr 471	. . . . . . . 8 |- ( ( A e. dom R1 /\ x e. A ) -> Ord A )
60		ordelsuc 6635	. . . . . . . 8 |- ( ( x e. A /\ Ord A ) -> ( x e. A <-> suc x C_ A ) )
61	56, 59, 60	syl2anc 671	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> ( x e. A <-> suc x C_ A ) )
62	56, 61	mpbid 215	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> suc x C_ A )
63	54, 19	sylib 201	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> suc x e. dom R1 )
64		simpl 463	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> A e. dom R1 )
65		r1ord3g 8237	. . . . . . 7 |- ( ( suc x e. dom R1 /\ A e. dom R1 ) -> ( suc x C_ A -> ( R1 ` suc x ) C_ ( R1 ` A ) ) )
66	63, 64, 65	syl2anc 671	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> ( suc x C_ A -> ( R1 ` suc x ) C_ ( R1 ` A ) ) )
67	62, 66	mpd 15	. . . . 5 |- ( ( A e. dom R1 /\ x e. A ) -> ( R1 ` suc x ) C_ ( R1 ` A ) )
68	55, 67	eqsstr3d 3435	. . . 4 |- ( ( A e. dom R1 /\ x e. A ) -> ~P ( R1 ` x ) C_ ( R1 ` A ) )
69	68	ralrimiva 2790	. . 3 |- ( A e. dom R1 -> A. x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
70		iunss 4289	. . 3 |- ( U_ x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) <-> A. x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
71	69, 70	sylibr 217	. 2 |- ( A e. dom R1 -> U_ x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
72	51, 71	eqssd 3417	1 |- ( A e. dom R1 -> ( R1 ` A ) = U_ x e. A ~P ( R1 ` x ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   \/ w3o 985   = wceq 1448   e. wcel 1891   A.wral 2737   E.wrex 2738   _Vcvv 3013   C_ wss 3372   (/)c0 3699   ~Pcpw 3919   U_ciun 4248   Tr wtr 4469   dom cdm 4812   Ord word 5401   Oncon0 5402   Lim wlim 5403   suc csuc 5404   Fun wfun 5555   ` cfv 5561   R1cr1 8220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-tr 4470  df-eprel 4723  df-id 4727  df-po 4733  df-so 4734  df-fr 4771  df-we 4773  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-pred 5359  df-ord 5405  df-on 5406  df-lim 5407  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-om 6681  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-r1 8222

This theorem is referenced by:  rankr1ai  8256  r1val3  8296