Theorem acdc5 8762

Description: A more general version of acdc 8764 that has an initial value and where the function F depends on k.

Hypothesis

Ref	Expression
acdc5.1	|- A e. _V

Assertion

Ref	Expression
acdc5	|- ((F:(NN X. A)-->(~PA \ {(/)}) /\ C e. A) -> E.g(g:NN-->A /\ (g` 1) = C /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))

Distinct variable groups:   g,k,A   g,F,k   C,g

Proof of Theorem acdc5

Step	Hyp	Ref	Expression
1		eqeq2 1893	. . . . . 6 |- (c = C -> ((g` 1) = c <-> (g` 1) = C))
2	1	3anbi2d 1173	. . . . 5 |- (c = C -> ((g:NN-->A /\ (g` 1) = c /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) <-> (g:NN-->A /\ (g` 1) = C /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)))))
3	2	exbidv 1657	. . . 4 |- (c = C -> (E.g(g:NN-->A /\ (g` 1) = c /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) <-> E.g(g:NN-->A /\ (g` 1) = C /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)))))
4	3	imbi2d 674	. . 3 |- (c = C -> ((F:(NN X. A)-->(~PA \ {(/)}) -> E.g(g:NN-->A /\ (g` 1) = c /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)))) <-> (F:(NN X. A)-->(~PA \ {(/)}) -> E.g(g:NN-->A /\ (g` 1) = C /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))))
5		acdc5.1	. . . . 5 |- A e. _V
6	5	weth 5949	. . . 4 |- E.r r We A
7		eleq1 1957	. . . . . . . . . . . 12 |- (a = x -> (a e. A <-> x e. A))
8		eleq1 1957	. . . . . . . . . . . 12 |- (b = y -> (b e. NN <-> y e. NN))
9	7, 8	bi2anan9 694	. . . . . . . . . . 11 |- ((a = x /\ b = y) -> ((a e. A /\ b e. NN) <-> (x e. A /\ y e. NN)))
10		opreq12 4891	. . . . . . . . . . . . . . . 16 |- ((b = y /\ a = x) -> (bFa) = (yFx))
11	10	ancoms 484	. . . . . . . . . . . . . . 15 |- ((a = x /\ b = y) -> (bFa) = (yFx))
12		rabeq 2289	. . . . . . . . . . . . . . . 16 |- ((bFa) = (yFx) -> {f e. (bFa) | A.h e. (bFa) -. hrf} = {f e. (yFx) | A.h e. (bFa) -. hrf})
13		raleq 2266	. . . . . . . . . . . . . . . . 17 |- ((bFa) = (yFx) -> (A.h e. (bFa) -. hrf <-> A.h e. (yFx) -. hrf))
14	13	rabbidv 2287	. . . . . . . . . . . . . . . 16 |- ((bFa) = (yFx) -> {f e. (yFx) | A.h e. (bFa) -. hrf} = {f e. (yFx) | A.h e. (yFx) -. hrf})
15	12, 14	eqtrd 1925	. . . . . . . . . . . . . . 15 |- ((bFa) = (yFx) -> {f e. (bFa) | A.h e. (bFa) -. hrf} = {f e. (yFx) | A.h e. (yFx) -. hrf})
16	11, 15	syl 12	. . . . . . . . . . . . . 14 |- ((a = x /\ b = y) -> {f e. (bFa) | A.h e. (bFa) -. hrf} = {f e. (yFx) | A.h e. (yFx) -. hrf})
17		breq2 3342	. . . . . . . . . . . . . . . . . 18 |- (f = v -> (hrf <-> hrv))
18	17	notbid 673	. . . . . . . . . . . . . . . . 17 |- (f = v -> (-. hrf <-> -. hrv))
19	18	ralbidv 2123	. . . . . . . . . . . . . . . 16 |- (f = v -> (A.h e. (yFx) -. hrf <-> A.h e. (yFx) -. hrv))
20		breq1 3341	. . . . . . . . . . . . . . . . . 18 |- (h = u -> (hrv <-> urv))
21	20	notbid 673	. . . . . . . . . . . . . . . . 17 |- (h = u -> (-. hrv <-> -. urv))
22	21	cbvralv 2280	. . . . . . . . . . . . . . . 16 |- (A.h e. (yFx) -. hrv <-> A.u e. (yFx) -. urv)
23	19, 22	syl6bb 595	. . . . . . . . . . . . . . 15 |- (f = v -> (A.h e. (yFx) -. hrf <-> A.u e. (yFx) -. urv))
24	23	cbvrabv 2422	. . . . . . . . . . . . . 14 |- {f e. (yFx) | A.h e. (yFx) -. hrf} = {v e. (yFx) | A.u e. (yFx) -. urv}
25	16, 24	syl6eq 1944	. . . . . . . . . . . . 13 |- ((a = x /\ b = y) -> {f e. (bFa) | A.h e. (bFa) -. hrf} = {v e. (yFx) | A.u e. (yFx) -. urv})
26	25	unieqd 3188	. . . . . . . . . . . 12 |- ((a = x /\ b = y) -> U.{f e. (bFa) | A.h e. (bFa) -. hrf} = U.{v e. (yFx) | A.u e. (yFx) -. urv})
27	26	eqeq2d 1895	. . . . . . . . . . 11 |- ((a = x /\ b = y) -> (d = U.{f e. (bFa) | A.h e. (bFa) -. hrf} <-> d = U.{v e. (yFx) | A.u e. (yFx) -. urv}))
28	9, 27	anbi12d 690	. . . . . . . . . 10 |- ((a = x /\ b = y) -> (((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf}) <-> ((x e. A /\ y e. NN) /\ d = U.{v e. (yFx) | A.u e. (yFx) -. urv})))
29	28	cbvoprab12v 4928	. . . . . . . . 9 |- {<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} = {<.<.x, y>., d>. | ((x e. A /\ y e. NN) /\ d = U.{v e. (yFx) | A.u e. (yFx) -. urv})}
30		eqeq1 1890	. . . . . . . . . . 11 |- (d = z -> (d = U.{v e. (yFx) | A.u e. (yFx) -. urv} <-> z = U.{v e. (yFx) | A.u e. (yFx) -. urv}))
31	30	anbi2d 678	. . . . . . . . . 10 |- (d = z -> (((x e. A /\ y e. NN) /\ d = U.{v e. (yFx) | A.u e. (yFx) -. urv}) <-> ((x e. A /\ y e. NN) /\ z = U.{v e. (yFx) | A.u e. (yFx) -. urv})))
32	31	cbvoprab3v 4929	. . . . . . . . 9 |- {<.<.x, y>., d>. | ((x e. A /\ y e. NN) /\ d = U.{v e. (yFx) | A.u e. (yFx) -. urv})} = {<.<.x, y>., z>. | ((x e. A /\ y e. NN) /\ z = U.{v e. (yFx) | A.u e. (yFx) -. urv})}
33	29, 32	eqtri 1908	. . . . . . . 8 |- {<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} = {<.<.x, y>., z>. | ((x e. A /\ y e. NN) /\ z = U.{v e. (yFx) | A.u e. (yFx) -. urv})}
34		eqid 1884	. . . . . . . 8 |- ({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))) = ({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))
35	5, 33, 34	acdc5lem2 8761	. . . . . . 7 |- ((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) -> (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))):NN-->A /\ (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` 1) = c /\ A.k e. NN (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` (k + 1)) e. ((k + 1)F(({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k))))
36		oprex 4907	. . . . . . . 8 |- ({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))) e. _V
37		feq1 4551	. . . . . . . . 9 |- (g = ({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))) -> (g:NN-->A <-> ({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))):NN-->A))
38		fveq1 4680	. . . . . . . . . 10 |- (g = ({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))) -> (g` 1) = (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` 1))
39	38	eqeq1d 1892	. . . . . . . . 9 |- (g = ({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))) -> ((g` 1) = c <-> (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` 1) = c))
40		fveq1 4680	. . . . . . . . . . 11 |- (g = ({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))) -> (g` (k + 1)) = (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` (k + 1)))
41		fveq1 4680	. . . . . . . . . . . 12 |- (g = ({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))) -> (g` k) = (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k))
42	41	opreq2d 4898	. . . . . . . . . . 11 |- (g = ({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))) -> ((k + 1)F(g` k)) = ((k + 1)F(({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k)))
43	40, 42	eleq12d 1965	. . . . . . . . . 10 |- (g = ({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))) -> ((g` (k + 1)) e. ((k + 1)F(g` k)) <-> (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` (k + 1)) e. ((k + 1)F(({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k))))
44	43	ralbidv 2123	. . . . . . . . 9 |- (g = ({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))) -> (A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)) <-> A.k e. NN (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` (k + 1)) e. ((k + 1)F(({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k))))
45	37, 39, 44	3anbi123d 1168	. . . . . . . 8 |- (g = ({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))) -> ((g:NN-->A /\ (g` 1) = c /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) <-> (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))):NN-->A /\ (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` 1) = c /\ A.k e. NN (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` (k + 1)) e. ((k + 1)F(({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k)))))
46	36, 45	cla4ev 2371	. . . . . . 7 |- ((({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1})))):NN-->A /\ (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` 1) = c /\ A.k e. NN (({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` (k + 1)) e. ((k + 1)F(({<.<.a, b>., d>. | ((a e. A /\ b e. NN) /\ d = U.{f e. (bFa) | A.h e. (bFa) -. hrf})} seq1 ({<.1, c>.} u. ( _I |` (NN \ {1}))))` k))) -> E.g(g:NN-->A /\ (g` 1) = c /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))
47	35, 46	syl 12	. . . . . 6 |- ((r We A /\ (c e. A /\ F:(NN X. A)-->(~PA \ {(/)}))) -> E.g(g:NN-->A /\ (g` 1) = c /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))
48	47	exp32 408	. . . . 5 |- (r We A -> (c e. A -> (F:(NN X. A)-->(~PA \ {(/)}) -> E.g(g:NN-->A /\ (g` 1) = c /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))))
49	48	19.23aiv 1674	. . . 4 |- (E.r r We A -> (c e. A -> (F:(NN X. A)-->(~PA \ {(/)}) -> E.g(g:NN-->A /\ (g` 1) = c /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))))))
50	6, 49	ax-mp 7	. . 3 |- (c e. A -> (F:(NN X. A)-->(~PA \ {(/)}) -> E.g(g:NN-->A /\ (g` 1) = c /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)))))
51	4, 50	vtoclga 2352	. 2 |- (C e. A -> (F:(NN X. A)-->(~PA \ {(/)}) -> E.g(g:NN-->A /\ (g` 1) = C /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)))))
52	51	impcom 378	1 |- ((F:(NN X. A)-->(~PA \ {(/)}) /\ C e. A) -> E.g(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   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  {crab 2108  _Vcvv 2292   \ cdif 2590   u. cun 2591  (/)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  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  1c1 6387   + caddc 6389  NNcn 6449   seq1 cseq1 7720

This theorem is referenced by:  bcthlem30 9306  acdc5g 15752

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  ax-ac 5906

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-iso 4015  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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