Theorem r1val1 8117

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 459	. . . . . 6 |- ( ( A e. dom R1 /\ A = (/) ) -> A = (/) )
2	1	fveq2d 5778	. . . . 5 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) = ( R1 ` (/) ) )
3		r10 8099	. . . . 5 |- ( R1 ` (/) ) = (/)
4	2, 3	syl6eq 2439	. . . 4 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) = (/) )
5		0ss 3741	. . . . 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 3451	. . 3 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
8		nfv 1715	. . . . 5 |- F/ x A e. dom R1
9		nfcv 2544	. . . . . 6 |- F/_ x ( R1 ` A )
10		nfiu1 4273	. . . . . 6 |- F/_ x U_ x e. A ~P ( R1 ` x )
11	9, 10	nfss 3410	. . . . 5 |- F/ x ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x )
12		simpr 459	. . . . . . . . . 10 |- ( ( A e. dom R1 /\ A = suc x ) -> A = suc x )
13	12	fveq2d 5778	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) = ( R1 ` suc x ) )
14		eleq1 2454	. . . . . . . . . . . 12 |- ( A = suc x -> ( A e. dom R1 <-> suc x e. dom R1 ) )
15	14	biimpac 484	. . . . . . . . . . 11 |- ( ( A e. dom R1 /\ A = suc x ) -> suc x e. dom R1 )
16		r1funlim 8097	. . . . . . . . . . . . 13 |- ( Fun R1 /\ Lim dom R1 )
17	16	simpri 460	. . . . . . . . . . . 12 |- Lim dom R1
18		limsuc 6583	. . . . . . . . . . . 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 212	. . . . . . . . . 10 |- ( ( A e. dom R1 /\ A = suc x ) -> x e. dom R1 )
21		r1sucg 8100	. . . . . . . . . 10 |- ( x e. dom R1 -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
22	20, 21	syl 16	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
23	13, 22	eqtrd 2423	. . . . . . . 8 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) = ~P ( R1 ` x ) )
24		vex 3037	. . . . . . . . . . 11 |- x e. _V
25	24	sucid 4871	. . . . . . . . . 10 |- x e. suc x
26	25, 12	syl5eleqr 2477	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> x e. A )
27		ssiun2 4286	. . . . . . . . 9 |- ( x e. A -> ~P ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
28	26, 27	syl 16	. . . . . . . 8 |- ( ( A e. dom R1 /\ A = suc x ) -> ~P ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
29	23, 28	eqsstrd 3451	. . . . . . 7 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
30	29	ex 432	. . . . . 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 2866	. . . 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 427	. . 3 |- ( ( A e. dom R1 /\ E. x e. On A = suc x ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
34		r1limg 8102	. . . . 5 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` A ) = U_ x e. A ( R1 ` x ) )
35		r1tr 8107	. . . . . . . . 9 |- Tr ( R1 ` x )
36		dftr4 4465	. . . . . . . . 9 |- ( Tr ( R1 ` x ) <-> ( R1 ` x ) C_ ~P ( R1 ` x ) )
37	35, 36	mpbi 208	. . . . . . . 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 2797	. . . . . 6 |- ( ( A e. dom R1 /\ Lim A ) -> A. x e. A ( R1 ` x ) C_ ~P ( R1 ` x ) )
40		ss2iun 4259	. . . . . 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 16	. . . . 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 3451	. . . 4 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
43	42	adantrl 713	. . 3 |- ( ( A e. dom R1 /\ ( A e. _V /\ Lim A ) ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
44		limord 4851	. . . . . . 7 |- ( Lim dom R1 -> Ord dom R1 )
45	17, 44	ax-mp 5	. . . . . 6 |- Ord dom R1
46		ordsson 6524	. . . . . 6 |- ( Ord dom R1 -> dom R1 C_ On )
47	45, 46	ax-mp 5	. . . . 5 |- dom R1 C_ On
48	47	sseli 3413	. . . 4 |- ( A e. dom R1 -> A e. On )
49		onzsl 6580	. . . 4 |- ( A e. On <-> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
50	48, 49	sylib 196	. . 3 |- ( A e. dom R1 -> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
51	7, 33, 43, 50	mpjao3dan 1293	. 2 |- ( A e. dom R1 -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
52		ordtr1 4835	. . . . . . . 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 451	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> x e. dom R1 )
55	54, 21	syl 16	. . . . 5 |- ( ( A e. dom R1 /\ x e. A ) -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
56		simpr 459	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> x e. A )
57		ordelord 4814	. . . . . . . . . 10 |- ( ( Ord dom R1 /\ A e. dom R1 ) -> Ord A )
58	45, 57	mpan 668	. . . . . . . . 9 |- ( A e. dom R1 -> Ord A )
59	58	adantr 463	. . . . . . . 8 |- ( ( A e. dom R1 /\ x e. A ) -> Ord A )
60		ordelsuc 6554	. . . . . . . 8 |- ( ( x e. A /\ Ord A ) -> ( x e. A <-> suc x C_ A ) )
61	56, 59, 60	syl2anc 659	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> ( x e. A <-> suc x C_ A ) )
62	56, 61	mpbid 210	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> suc x C_ A )
63	54, 19	sylib 196	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> suc x e. dom R1 )
64		simpl 455	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> A e. dom R1 )
65		r1ord3g 8110	. . . . . . 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 659	. . . . . 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 3452	. . . 4 |- ( ( A e. dom R1 /\ x e. A ) -> ~P ( R1 ` x ) C_ ( R1 ` A ) )
69	68	ralrimiva 2796	. . 3 |- ( A e. dom R1 -> A. x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
70		iunss 4284	. . 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 212	. 2 |- ( A e. dom R1 -> U_ x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
72	51, 71	eqssd 3434	1 |- ( A e. dom R1 -> ( R1 ` A ) = U_ x e. A ~P ( R1 ` x ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   \/ w3o 970   = wceq 1399   e. wcel 1826   A.wral 2732   E.wrex 2733   _Vcvv 3034   C_ wss 3389   (/)c0 3711   ~Pcpw 3927   U_ciun 4243   Tr wtr 4460   Ord word 4791   Oncon0 4792   Lim wlim 4793   suc csuc 4794   dom cdm 4913   Fun wfun 5490   ` cfv 5496   R1cr1 8093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-om 6600  df-recs 6960  df-rdg 6994  df-r1 8095

This theorem is referenced by:  rankr1ai  8129  r1val3  8169