Theorem r1val1 8649

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 477	. . . . . 6 |- ( ( A e. dom R1 /\ A = (/) ) -> A = (/) )
2	1	fveq2d 6195	. . . . 5 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) = ( R1 ` (/) ) )
3		r10 8631	. . . . 5 |- ( R1 ` (/) ) = (/)
4	2, 3	syl6eq 2672	. . . 4 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) = (/) )
5		0ss 3972	. . . . 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 3639	. . 3 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
8		nfv 1843	. . . . 5 |- F/ x A e. dom R1
9		nfcv 2764	. . . . . 6 |- F/_ x ( R1 ` A )
10		nfiu1 4550	. . . . . 6 |- F/_ x U_ x e. A ~P ( R1 ` x )
11	9, 10	nfss 3596	. . . . 5 |- F/ x ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x )
12		simpr 477	. . . . . . . . . 10 |- ( ( A e. dom R1 /\ A = suc x ) -> A = suc x )
13	12	fveq2d 6195	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) = ( R1 ` suc x ) )
14		eleq1 2689	. . . . . . . . . . . 12 |- ( A = suc x -> ( A e. dom R1 <-> suc x e. dom R1 ) )
15	14	biimpac 503	. . . . . . . . . . 11 |- ( ( A e. dom R1 /\ A = suc x ) -> suc x e. dom R1 )
16		r1funlim 8629	. . . . . . . . . . . . 13 |- ( Fun R1 /\ Lim dom R1 )
17	16	simpri 478	. . . . . . . . . . . 12 |- Lim dom R1
18		limsuc 7049	. . . . . . . . . . . 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 224	. . . . . . . . . 10 |- ( ( A e. dom R1 /\ A = suc x ) -> x e. dom R1 )
21		r1sucg 8632	. . . . . . . . . 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 2656	. . . . . . . 8 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) = ~P ( R1 ` x ) )
24		vex 3203	. . . . . . . . . . 11 |- x e. _V
25	24	sucid 5804	. . . . . . . . . 10 |- x e. suc x
26	25, 12	syl5eleqr 2708	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> x e. A )
27		ssiun2 4563	. . . . . . . . 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 3639	. . . . . . 7 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
30	29	ex 450	. . . . . 6 |- ( A e. dom R1 -> ( A = suc x -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) ) )
31	30	a1d 25	. . . . 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 3026	. . . 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 445	. . 3 |- ( ( A e. dom R1 /\ E. x e. On A = suc x ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
34		r1limg 8634	. . . . 5 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` A ) = U_ x e. A ( R1 ` x ) )
35		r1tr 8639	. . . . . . . . 9 |- Tr ( R1 ` x )
36		dftr4 4757	. . . . . . . . 9 |- ( Tr ( R1 ` x ) <-> ( R1 ` x ) C_ ~P ( R1 ` x ) )
37	35, 36	mpbi 220	. . . . . . . 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 2967	. . . . . 6 |- ( ( A e. dom R1 /\ Lim A ) -> A. x e. A ( R1 ` x ) C_ ~P ( R1 ` x ) )
40		ss2iun 4536	. . . . . 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 3639	. . . 4 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
43	42	adantrl 752	. . 3 |- ( ( A e. dom R1 /\ ( A e. _V /\ Lim A ) ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
44		limord 5784	. . . . . . 7 |- ( Lim dom R1 -> Ord dom R1 )
45	17, 44	ax-mp 5	. . . . . 6 |- Ord dom R1
46		ordsson 6989	. . . . . 6 |- ( Ord dom R1 -> dom R1 C_ On )
47	45, 46	ax-mp 5	. . . . 5 |- dom R1 C_ On
48	47	sseli 3599	. . . 4 |- ( A e. dom R1 -> A e. On )
49		onzsl 7046	. . . 4 |- ( A e. On <-> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
50	48, 49	sylib 208	. . 3 |- ( A e. dom R1 -> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
51	7, 33, 43, 50	mpjao3dan 1395	. 2 |- ( A e. dom R1 -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
52		ordtr1 5767	. . . . . . . 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 469	. . . . . 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 477	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> x e. A )
57		ordelord 5745	. . . . . . . . . 10 |- ( ( Ord dom R1 /\ A e. dom R1 ) -> Ord A )
58	45, 57	mpan 706	. . . . . . . . 9 |- ( A e. dom R1 -> Ord A )
59	58	adantr 481	. . . . . . . 8 |- ( ( A e. dom R1 /\ x e. A ) -> Ord A )
60		ordelsuc 7020	. . . . . . . 8 |- ( ( x e. A /\ Ord A ) -> ( x e. A <-> suc x C_ A ) )
61	56, 59, 60	syl2anc 693	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> ( x e. A <-> suc x C_ A ) )
62	56, 61	mpbid 222	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> suc x C_ A )
63	54, 19	sylib 208	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> suc x e. dom R1 )
64		simpl 473	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> A e. dom R1 )
65		r1ord3g 8642	. . . . . . 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 693	. . . . . 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 3640	. . . 4 |- ( ( A e. dom R1 /\ x e. A ) -> ~P ( R1 ` x ) C_ ( R1 ` A ) )
69	68	ralrimiva 2966	. . 3 |- ( A e. dom R1 -> A. x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
70		iunss 4561	. . 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 224	. 2 |- ( A e. dom R1 -> U_ x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
72	51, 71	eqssd 3620	1 |- ( A e. dom R1 -> ( R1 ` A ) = U_ x e. A ~P ( R1 ` x ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U_ciun 4520   Tr wtr 4752   dom cdm 5114   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725   Fun wfun 5882   ` cfv 5888   R1cr1 8625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627

This theorem is referenced by:  rankr1ai  8661  r1val3  8701