Theorem acdc5lem2 8761

Description: Lemma for acdc5 8762. Build a sequence G starting at value c, as follows. Given a previous value x of G, we construct, for the next value of G, the v such that A.u e. (yFx)-. urv, which is unique when r is a well-ordering on A.

Hypotheses

Ref	Expression
acdc5lem.1	|- A e. _V
acdc5lem.2	|- S = {<.<.x, y>., z>. | ((x e. A /\ y e. NN) /\ z = U.{v e. (yFx) | A.u e. (yFx) -. urv})}
acdc5lem.3	|- G = (S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))

Assertion

Ref	Expression
acdc5lem2	|- ((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) -> (G:NN-->A /\ (G` 1) = c /\ A.k e. NN (G` (k + 1)) e. ((k + 1)F(G` k))))

Distinct variable groups:   u,k,v,x,y,z,A   k,F,u,v,x,y,z   u,G,v,x,y,z   k,c,x,y,z   k,r,u,v,x,y,z

Proof of Theorem acdc5lem2

Step	Hyp	Ref	Expression
1		id 73	. . . . . . 7 |- (c e. A -> c e. A)
2		1nn 7117	. . . . . . . . 9 |- 1 e. NN
3	2	elisseti 2301	. . . . . . . 8 |- 1 e. _V
4		visset 2295	. . . . . . . 8 |- c e. _V
5		eqid 1884	. . . . . . . 8 |- ({<.1, c>.} u. ( _I |` (NN \ {1}))) = ({<.1, c>.} u. ( _I |` (NN \ {1})))
6	3, 4, 5	fvsnun1 4764	. . . . . . 7 |- (({<.1, c>.} u. ( _I |` (NN \ {1})))` 1) = c
7	1, 6	syl5eqel 1975	. . . . . 6 |- (c e. A -> (({<.1, c>.} u. ( _I |` (NN \ {1})))` 1) e. A)
8	7	ad2antrl 442	. . . . 5 |- ((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) -> (({<.1, c>.} u. ( _I |` (NN \ {1})))` 1) e. A)
9		f1oi 4671	. . . . . . . 8 |- ( _I |` (NN \ {1})):(NN \ {1})-1-1-onto->(NN \ {1})
10		f1of 4635	. . . . . . . 8 |- (( _I |` (NN \ {1})):(NN \ {1})-1-1-onto->(NN \ {1}) -> ( _I |` (NN \ {1})):(NN \ {1})-->(NN \ {1}))
11	9, 10	ax-mp 7	. . . . . . 7 |- ( _I |` (NN \ {1})):(NN \ {1})-->(NN \ {1})
12		difss 2735	. . . . . . 7 |- (NN \ {1}) C_ NN
13		fss 4571	. . . . . . 7 |- ((( _I |` (NN \ {1})):(NN \ {1})-->(NN \ {1}) /\ (NN \ {1}) C_ NN) -> ( _I |` (NN \ {1})):(NN \ {1})-->NN)
14	11, 12, 13	mp2an 761	. . . . . 6 |- ( _I |` (NN \ {1})):(NN \ {1})-->NN
15		resundir 4230	. . . . . . . 8 |- (({<.1, c>.} u. ( _I |` (NN \ {1}))) |` (NN \ {1})) = (({<.1, c>.} |` (NN \ {1})) u. (( _I |` (NN \ {1})) |` (NN \ {1})))
16		difdisj 2945	. . . . . . . . . 10 |- ({1} i^i (NN \ {1})) = (/)
17	3, 4	f1osn 4674	. . . . . . . . . . . 12 |- {<.1, c>.}:{1}-1-1-onto->{c}
18		f1ofn 4636	. . . . . . . . . . . 12 |- ({<.1, c>.}:{1}-1-1-onto->{c} -> {<.1, c>.} Fn {1})
19	17, 18	ax-mp 7	. . . . . . . . . . 11 |- {<.1, c>.} Fn {1}
20		fnresdisj 4523	. . . . . . . . . . 11 |- ({<.1, c>.} Fn {1} -> (({1} i^i (NN \ {1})) = (/) <-> ({<.1, c>.} |` (NN \ {1})) = (/)))
21	19, 20	ax-mp 7	. . . . . . . . . 10 |- (({1} i^i (NN \ {1})) = (/) <-> ({<.1, c>.} |` (NN \ {1})) = (/))
22	16, 21	mpbi 206	. . . . . . . . 9 |- ({<.1, c>.} |` (NN \ {1})) = (/)
23		residm 4246	. . . . . . . . 9 |- (( _I |` (NN \ {1})) |` (NN \ {1})) = ( _I |` (NN \ {1}))
24	22, 23	uneq12i 2753	. . . . . . . 8 |- (({<.1, c>.} |` (NN \ {1})) u. (( _I |` (NN \ {1})) |` (NN \ {1}))) = ((/) u. ( _I |` (NN \ {1})))
25		uncom 2744	. . . . . . . . 9 |- (( _I |` (NN \ {1})) u. (/)) = ((/) u. ( _I |` (NN \ {1})))
26		un0 2896	. . . . . . . . 9 |- (( _I |` (NN \ {1})) u. (/)) = ( _I |` (NN \ {1}))
27	25, 26	eqtr3i 1910	. . . . . . . 8 |- ((/) u. ( _I |` (NN \ {1}))) = ( _I |` (NN \ {1}))
28	15, 24, 27	3eqtri 1912	. . . . . . 7 |- (({<.1, c>.} u. ( _I |` (NN \ {1}))) |` (NN \ {1})) = ( _I |` (NN \ {1}))
29	28	feq1i 4558	. . . . . 6 |- ((({<.1, c>.} u. ( _I |` (NN \ {1}))) |` (NN \ {1})):(NN \ {1})-->NN <-> ( _I |` (NN \ {1})):(NN \ {1})-->NN)
30	14, 29	mpbir 207	. . . . 5 |- (({<.1, c>.} u. ( _I |` (NN \ {1}))) |` (NN \ {1})):(NN \ {1})-->NN
31	8, 30	jctir 317	. . . 4 |- ((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) -> ((({<.1, c>.} u. ( _I |` (NN \ {1})))` 1) e. A /\ (({<.1, c>.} u. ( _I |` (NN \ {1}))) |` (NN \ {1})):(NN \ {1})-->NN))
32		acdc5lem.1	. . . . . . . . . . 11 |- A e. _V
33		acdc5lem.2	. . . . . . . . . . 11 |- S = {<.<.x, y>., z>. | ((x e. A /\ y e. NN) /\ z = U.{v e. (yFx) | A.u e. (yFx) -. urv})}
34		acdc5lem.3	. . . . . . . . . . 11 |- G = (S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))
35	32, 33, 34	acdc5lem1 8760	. . . . . . . . . 10 |- (((r We A /\ F:(NN X. A)-->(~PA \ {(/)})) /\ (s e. A /\ t e. NN)) -> ((sSt) e. (tFs) /\ (sSt) e. A))
36	35	simprd 352	. . . . . . . . 9 |- (((r We A /\ F:(NN X. A)-->(~PA \ {(/)})) /\ (s e. A /\ t e. NN)) -> (sSt) e. A)
37	36	ex 402	. . . . . . . 8 |- ((r We A /\ F:(NN X. A)-->(~PA \ {(/)})) -> ((s e. A /\ t e. NN) -> (sSt) e. A))
38	37	r19.21aivv 2183	. . . . . . 7 |- ((r We A /\ F:(NN X. A)-->(~PA \ {(/)})) -> A.s e. A A.t e. NN (sSt) e. A)
39		oprex 4907	. . . . . . . . . 10 |- (yFx) e. _V
40	39	rabex 3461	. . . . . . . . 9 |- {v e. (yFx) | A.u e. (yFx) -. urv} e. _V
41	40	uniex 3794	. . . . . . . 8 |- U.{v e. (yFx) | A.u e. (yFx) -. urv} e. _V
42	41, 33	fnoprab2 5064	. . . . . . 7 |- S Fn (A X. NN)
43	38, 42	jctil 316	. . . . . 6 |- ((r We A /\ F:(NN X. A)-->(~PA \ {(/)})) -> (S Fn (A X. NN) /\ A.s e. A A.t e. NN (sSt) e. A))
44		ffnoprv 4943	. . . . . 6 |- (S:(A X. NN)-->A <-> (S Fn (A X. NN) /\ A.s e. A A.t e. NN (sSt) e. A))
45	43, 44	sylibr 217	. . . . 5 |- ((r We A /\ F:(NN X. A)-->(~PA \ {(/)})) -> S:(A X. NN)-->A)
46	45	adantrl 430	. . . 4 |- ((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) -> S:(A X. NN)-->A)
47		nnex 7116	. . . . . . 7 |- NN e. _V
48	32, 47, 33	oprabex2 4950	. . . . . 6 |- S e. _V
49		snex 3492	. . . . . . 7 |- {<.1, c>.} e. _V
50		difexg 3458	. . . . . . . . 9 |- (NN e. _V -> (NN \ {1}) e. _V)
51	47, 50	ax-mp 7	. . . . . . . 8 |- (NN \ {1}) e. _V
52		resiexg 4253	. . . . . . . 8 |- ((NN \ {1}) e. _V -> ( _I |` (NN \ {1})) e. _V)
53	51, 52	ax-mp 7	. . . . . . 7 |- ( _I |` (NN \ {1})) e. _V
54	49, 53	unex 3796	. . . . . 6 |- ({<.1, c>.} u. ( _I |` (NN \ {1}))) e. _V
55	48, 54	seq1f2 7737	. . . . 5 |- (((({<.1, c>.} u. ( _I |` (NN \ {1})))` 1) e. A /\ (({<.1, c>.} u. ( _I |` (NN \ {1}))) |` (NN \ {1})):(NN \ {1})-->NN /\ S:(A X. NN)-->A) -> (S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))):NN-->A)
56	55	3expa 1067	. . . 4 |- ((((({<.1, c>.} u. ( _I |` (NN \ {1})))` 1) e. A /\ (({<.1, c>.} u. ( _I |` (NN \ {1}))) |` (NN \ {1})):(NN \ {1})-->NN) /\ S:(A X. NN)-->A) -> (S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))):NN-->A)
57	31, 46, 56	syl11anc 524	. . 3 |- ((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) -> (S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))):NN-->A)
58	34	feq1i 4558	. . 3 |- (G:NN-->A <-> (S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))):NN-->A)
59	57, 58	sylibr 217	. 2 |- ((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) -> G:NN-->A)
60	34	fveq1i 4682	. . . 4 |- (G` 1) = ((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` 1)
61	48, 54	seq11 7730	. . . 4 |- ((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` 1) = (({<.1, c>.} u. ( _I |` (NN \ {1})))` 1)
62	60, 61, 6	3eqtri 1912	. . 3 |- (G` 1) = c
63	62	a1i 8	. 2 |- ((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) -> (G` 1) = c)
64	48, 54	seq1p1 7731	. . . . . . 7 |- (k e. NN -> ((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` (k + 1)) = (((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k)S(({<.1, c>.} u. ( _I |` (NN \ {1})))` (k + 1))))
65		seq1lem2 7723	. . . . . . . . . 10 |- (k e. NN -> (k + 1) e. (NN \ {1}))
66	3, 4, 5	fvsnun2 4765	. . . . . . . . . 10 |- ((k + 1) e. (NN \ {1}) -> (({<.1, c>.} u. ( _I |` (NN \ {1})))` (k + 1)) = ( _I ` (k + 1)))
67	65, 66	syl 12	. . . . . . . . 9 |- (k e. NN -> (({<.1, c>.} u. ( _I |` (NN \ {1})))` (k + 1)) = ( _I ` (k + 1)))
68		oprex 4907	. . . . . . . . . 10 |- (k + 1) e. _V
69		fvi 4818	. . . . . . . . . 10 |- ((k + 1) e. _V -> ( _I ` (k + 1)) = (k + 1))
70	68, 69	ax-mp 7	. . . . . . . . 9 |- ( _I ` (k + 1)) = (k + 1)
71	67, 70	syl6eq 1944	. . . . . . . 8 |- (k e. NN -> (({<.1, c>.} u. ( _I |` (NN \ {1})))` (k + 1)) = (k + 1))
72	71	opreq2d 4898	. . . . . . 7 |- (k e. NN -> (((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k)S(({<.1, c>.} u. ( _I |` (NN \ {1})))` (k + 1))) = (((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k)S(k + 1)))
73	64, 72	eqtrd 1925	. . . . . 6 |- (k e. NN -> ((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` (k + 1)) = (((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k)S(k + 1)))
74	34	fveq1i 4682	. . . . . 6 |- (G` (k + 1)) = ((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` (k + 1))
75	34	fveq1i 4682	. . . . . . 7 |- (G` k) = ((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k)
76	75	opreq1i 4892	. . . . . 6 |- ((G` k)S(k + 1)) = (((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k)S(k + 1))
77	73, 74, 76	3eqtr4g 1953	. . . . 5 |- (k e. NN -> (G` (k + 1)) = ((G` k)S(k + 1)))
78	77	adantl 424	. . . 4 |- (((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) /\ k e. NN) -> (G` (k + 1)) = ((G` k)S(k + 1)))
79		simpll 448	. . . . 5 |- (((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) /\ k e. NN) -> r We A)
80		simplrr 455	. . . . 5 |- (((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) /\ k e. NN) -> F:(NN X. A)-->(~PA \ {(/)}))
81		simpr 350	. . . . . . 7 |- (((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) /\ k e. NN) -> k e. NN)
82		simplrl 454	. . . . . . 7 |- (((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) /\ k e. NN) -> c e. A)
83	46	adantr 425	. . . . . . 7 |- (((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) /\ k e. NN) -> S:(A X. NN)-->A)
84		simp2 877	. . . . . . . . 9 |- ((k e. NN /\ c e. A /\ S:(A X. NN)-->A) -> c e. A)
85	84, 6	syl5eqel 1975	. . . . . . . 8 |- ((k e. NN /\ c e. A /\ S:(A X. NN)-->A) -> (({<.1, c>.} u. ( _I |` (NN \ {1})))` 1) e. A)
86	30	a1i 8	. . . . . . . 8 |- ((k e. NN /\ c e. A /\ S:(A X. NN)-->A) -> (({<.1, c>.} u. ( _I |` (NN \ {1}))) |` (NN \ {1})):(NN \ {1})-->NN)
87		simp3 878	. . . . . . . 8 |- ((k e. NN /\ c e. A /\ S:(A X. NN)-->A) -> S:(A X. NN)-->A)
88		simp1 876	. . . . . . . 8 |- ((k e. NN /\ c e. A /\ S:(A X. NN)-->A) -> k e. NN)
89	48, 54	seq1cl2 7739	. . . . . . . 8 |- ((((({<.1, c>.} u. ( _I |` (NN \ {1})))` 1) e. A /\ (({<.1, c>.} u. ( _I |` (NN \ {1}))) |` (NN \ {1})):(NN \ {1})-->NN /\ S:(A X. NN)-->A) /\ k e. NN) -> ((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k) e. A)
90	85, 86, 87, 88, 89	syl31anc 1103	. . . . . . 7 |- ((k e. NN /\ c e. A /\ S:(A X. NN)-->A) -> ((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k) e. A)
91	81, 82, 83, 90	syl111anc 1100	. . . . . 6 |- (((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) /\ k e. NN) -> ((S seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k) e. A)
92	91, 75	syl5eqel 1975	. . . . 5 |- (((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) /\ k e. NN) -> (G` k) e. A)
93		peano2nn 7118	. . . . . 6 |- (k e. NN -> (k + 1) e. NN)
94	93	adantl 424	. . . . 5 |- (((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) /\ k e. NN) -> (k + 1) e. NN)
95	32, 33, 34	acdc5lem1 8760	. . . . . 6 |- (((r We A /\ F:(NN X. A)-->(~PA \ {(/)})) /\ ((G` k) e. A /\ (k + 1) e. NN)) -> (((G` k)S(k + 1)) e. ((k + 1)F(G` k)) /\ ((G` k)S(k + 1)) e. A))
96	95	simplld 348	. . . . 5 |- (((r We A /\ F:(NN X. A)-->(~PA \ {(/)})) /\ ((G` k) e. A /\ (k + 1) e. NN)) -> ((G` k)S(k + 1)) e. ((k + 1)F(G` k)))
97	79, 80, 92, 94, 96	syl22anc 1101	. . . 4 |- (((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) /\ k e. NN) -> ((G` k)S(k + 1)) e. ((k + 1)F(G` k)))
98	78, 97	eqeltrd 1971	. . 3 |- (((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) /\ k e. NN) -> (G` (k + 1)) e. ((k + 1)F(G` k)))
99	98	r19.21aiva 2176	. 2 |- ((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) -> A.k e. NN (G` (k + 1)) e. ((k + 1)F(G` k)))
100	59, 63, 99	3jca 1050	1 |- ((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) -> (G:NN-->A /\ (G` 1) = c /\ A.k e. NN (G` (k + 1)) e. ((k + 1)F(G` k))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108  _Vcvv 2292   \ cdif 2590   u. cun 2591   i^i cin 2592   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  {csn 3044  <.cop 3046  U.cuni 3177   class class class wbr 3338   _I cid 3582   We wwe 3624   X. cxp 3984   |` cres 3988   Fn wfn 3993  -->wf 3994  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  {copab2 4885  1c1 6387   + caddc 6389  NNcn 6449   seq1 cseq1 7720

This theorem is referenced by:  acdc5 8762

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-seq1 7721

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