Theorem r1val1 8195

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		r1funlim 8175	. . . . . . . 8 |- ( Fun R1 /\ Lim dom R1 )
2	1	simpri 462	. . . . . . 7 |- Lim dom R1
3		limord 4932	. . . . . . 7 |- ( Lim dom R1 -> Ord dom R1 )
4	2, 3	ax-mp 5	. . . . . 6 |- Ord dom R1
5		ordsson 6598	. . . . . 6 |- ( Ord dom R1 -> dom R1 C_ On )
6	4, 5	ax-mp 5	. . . . 5 |- dom R1 C_ On
7	6	sseli 3495	. . . 4 |- ( A e. dom R1 -> A e. On )
8		onzsl 6654	. . . 4 |- ( A e. On <-> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
9	7, 8	sylib 196	. . 3 |- ( A e. dom R1 -> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
10		simpr 461	. . . . . . 7 |- ( ( A e. dom R1 /\ A = (/) ) -> A = (/) )
11	10	fveq2d 5863	. . . . . 6 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) = ( R1 ` (/) ) )
12		r10 8177	. . . . . 6 |- ( R1 ` (/) ) = (/)
13	11, 12	syl6eq 2519	. . . . 5 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) = (/) )
14		0ss 3809	. . . . . 6 |- (/) C_ U_ x e. A ~P ( R1 ` x )
15	14	a1i 11	. . . . 5 |- ( ( A e. dom R1 /\ A = (/) ) -> (/) C_ U_ x e. A ~P ( R1 ` x ) )
16	13, 15	eqsstrd 3533	. . . 4 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
17		nfv 1678	. . . . . 6 |- F/ x A e. dom R1
18		nfcv 2624	. . . . . . 7 |- F/_ x ( R1 ` A )
19		nfiu1 4350	. . . . . . 7 |- F/_ x U_ x e. A ~P ( R1 ` x )
20	18, 19	nfss 3492	. . . . . 6 |- F/ x ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x )
21		simpr 461	. . . . . . . . . . 11 |- ( ( A e. dom R1 /\ A = suc x ) -> A = suc x )
22	21	fveq2d 5863	. . . . . . . . . 10 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) = ( R1 ` suc x ) )
23		eleq1 2534	. . . . . . . . . . . . 13 |- ( A = suc x -> ( A e. dom R1 <-> suc x e. dom R1 ) )
24	23	biimpac 486	. . . . . . . . . . . 12 |- ( ( A e. dom R1 /\ A = suc x ) -> suc x e. dom R1 )
25		limsuc 6657	. . . . . . . . . . . . 13 |- ( Lim dom R1 -> ( x e. dom R1 <-> suc x e. dom R1 ) )
26	2, 25	ax-mp 5	. . . . . . . . . . . 12 |- ( x e. dom R1 <-> suc x e. dom R1 )
27	24, 26	sylibr 212	. . . . . . . . . . 11 |- ( ( A e. dom R1 /\ A = suc x ) -> x e. dom R1 )
28		r1sucg 8178	. . . . . . . . . . 11 |- ( x e. dom R1 -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
29	27, 28	syl 16	. . . . . . . . . 10 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
30	22, 29	eqtrd 2503	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) = ~P ( R1 ` x ) )
31		vex 3111	. . . . . . . . . . . 12 |- x e. _V
32	31	sucid 4952	. . . . . . . . . . 11 |- x e. suc x
33	32, 21	syl5eleqr 2557	. . . . . . . . . 10 |- ( ( A e. dom R1 /\ A = suc x ) -> x e. A )
34		ssiun2 4363	. . . . . . . . . 10 |- ( x e. A -> ~P ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
35	33, 34	syl 16	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> ~P ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
36	30, 35	eqsstrd 3533	. . . . . . . 8 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
37	36	ex 434	. . . . . . 7 |- ( A e. dom R1 -> ( A = suc x -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) ) )
38	37	a1d 25	. . . . . 6 |- ( A e. dom R1 -> ( x e. On -> ( A = suc x -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) ) ) )
39	17, 20, 38	rexlimd 2942	. . . . 5 |- ( A e. dom R1 -> ( E. x e. On A = suc x -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) ) )
40	39	imp 429	. . . 4 |- ( ( A e. dom R1 /\ E. x e. On A = suc x ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
41		r1limg 8180	. . . . . 6 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` A ) = U_ x e. A ( R1 ` x ) )
42		r1tr 8185	. . . . . . . . . 10 |- Tr ( R1 ` x )
43		dftr4 4540	. . . . . . . . . 10 |- ( Tr ( R1 ` x ) <-> ( R1 ` x ) C_ ~P ( R1 ` x ) )
44	42, 43	mpbi 208	. . . . . . . . 9 |- ( R1 ` x ) C_ ~P ( R1 ` x )
45	44	a1i 11	. . . . . . . 8 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` x ) C_ ~P ( R1 ` x ) )
46	45	ralrimivw 2874	. . . . . . 7 |- ( ( A e. dom R1 /\ Lim A ) -> A. x e. A ( R1 ` x ) C_ ~P ( R1 ` x ) )
47		ss2iun 4336	. . . . . . 7 |- ( A. x e. A ( R1 ` x ) C_ ~P ( R1 ` x ) -> U_ x e. A ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
48	46, 47	syl 16	. . . . . 6 |- ( ( A e. dom R1 /\ Lim A ) -> U_ x e. A ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
49	41, 48	eqsstrd 3533	. . . . 5 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
50	49	adantrl 715	. . . 4 |- ( ( A e. dom R1 /\ ( A e. _V /\ Lim A ) ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
51	16, 40, 50	3jaodan 1289	. . 3 |- ( ( A e. dom R1 /\ ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
52	9, 51	mpdan 668	. 2 |- ( A e. dom R1 -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
53		ordtr1 4916	. . . . . . . 8 |- ( Ord dom R1 -> ( ( x e. A /\ A e. dom R1 ) -> x e. dom R1 ) )
54	4, 53	ax-mp 5	. . . . . . 7 |- ( ( x e. A /\ A e. dom R1 ) -> x e. dom R1 )
55	54	ancoms 453	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> x e. dom R1 )
56	55, 28	syl 16	. . . . 5 |- ( ( A e. dom R1 /\ x e. A ) -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
57		simpr 461	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> x e. A )
58		ordelord 4895	. . . . . . . . . 10 |- ( ( Ord dom R1 /\ A e. dom R1 ) -> Ord A )
59	4, 58	mpan 670	. . . . . . . . 9 |- ( A e. dom R1 -> Ord A )
60	59	adantr 465	. . . . . . . 8 |- ( ( A e. dom R1 /\ x e. A ) -> Ord A )
61		ordelsuc 6628	. . . . . . . 8 |- ( ( x e. A /\ Ord A ) -> ( x e. A <-> suc x C_ A ) )
62	57, 60, 61	syl2anc 661	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> ( x e. A <-> suc x C_ A ) )
63	57, 62	mpbid 210	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> suc x C_ A )
64	55, 26	sylib 196	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> suc x e. dom R1 )
65		simpl 457	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> A e. dom R1 )
66		r1ord3g 8188	. . . . . . 7 |- ( ( suc x e. dom R1 /\ A e. dom R1 ) -> ( suc x C_ A -> ( R1 ` suc x ) C_ ( R1 ` A ) ) )
67	64, 65, 66	syl2anc 661	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> ( suc x C_ A -> ( R1 ` suc x ) C_ ( R1 ` A ) ) )
68	63, 67	mpd 15	. . . . 5 |- ( ( A e. dom R1 /\ x e. A ) -> ( R1 ` suc x ) C_ ( R1 ` A ) )
69	56, 68	eqsstr3d 3534	. . . 4 |- ( ( A e. dom R1 /\ x e. A ) -> ~P ( R1 ` x ) C_ ( R1 ` A ) )
70	69	ralrimiva 2873	. . 3 |- ( A e. dom R1 -> A. x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
71		iunss 4361	. . 3 |- ( U_ x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) <-> A. x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
72	70, 71	sylibr 212	. 2 |- ( A e. dom R1 -> U_ x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
73	52, 72	eqssd 3516	1 |- ( A e. dom R1 -> ( R1 ` A ) = U_ x e. A ~P ( R1 ` x ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   \/ w3o 967   = wceq 1374   e. wcel 1762   A.wral 2809   E.wrex 2810   _Vcvv 3108   C_ wss 3471   (/)c0 3780   ~Pcpw 4005   U_ciun 4320   Tr wtr 4535   Ord word 4872   Oncon0 4873   Lim wlim 4874   suc csuc 4875   dom cdm 4994   Fun wfun 5575   ` cfv 5581   R1cr1 8171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173

This theorem is referenced by:  rankr1ai  8207  r1val3  8247