Theorem acdcALT 8765

Description: Dependent Choice. Axiom DC1 of [Schechter] p. 149. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence.

Hypothesis

Ref	Expression
acdcALT.1	|- A e. _V

Assertion

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

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

Proof of Theorem acdcALT

Step	Hyp	Ref	Expression
1		acdcALT.1	. . . 4 |- A e. _V
2	1	acdc2 8759	. . 3 |- ((A =/= (/) /\ {<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))}:(NN X. A)-->(~PA \ {(/)})) -> E.g(g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1){<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))} (g` k))))
3		ffvelrn 4787	. . . . . . 7 |- ((F:A-->(~PA \ {(/)}) /\ y e. A) -> (F` y) e. (~PA \ {(/)}))
4	3	ex 402	. . . . . 6 |- (F:A-->(~PA \ {(/)}) -> (y e. A -> (F` y) e. (~PA \ {(/)})))
5	4	adantld 426	. . . . 5 |- (F:A-->(~PA \ {(/)}) -> ((x e. NN /\ y e. A) -> (F` y) e. (~PA \ {(/)})))
6	5	r19.21aivv 2183	. . . 4 |- (F:A-->(~PA \ {(/)}) -> A.x e. NN A.y e. A (F` y) e. (~PA \ {(/)}))
7		eqid 1884	. . . . 5 |- {<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))} = {<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))}
8	7	foprab2 5061	. . . 4 |- (A.x e. NN A.y e. A (F` y) e. (~PA \ {(/)}) <-> {<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))}:(NN X. A)-->(~PA \ {(/)}))
9	6, 8	sylib 215	. . 3 |- (F:A-->(~PA \ {(/)}) -> {<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))}:(NN X. A)-->(~PA \ {(/)}))
10	2, 9	sylan2 500	. 2 |- ((A =/= (/) /\ F:A-->(~PA \ {(/)})) -> E.g(g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1){<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))} (g` k))))
11		peano2nn 7118	. . . . . . . 8 |- (k e. NN -> (k + 1) e. NN)
12	11	adantl 424	. . . . . . 7 |- ((g:NN-->A /\ k e. NN) -> (k + 1) e. NN)
13		ffvelrn 4787	. . . . . . 7 |- ((g:NN-->A /\ k e. NN) -> (g` k) e. A)
14		fvex 4689	. . . . . . . 8 |- (F` (g` k)) e. _V
15		eqidd 1885	. . . . . . . 8 |- (x = (k + 1) -> (F` y) = (F` y))
16		fveq2 4681	. . . . . . . 8 |- (y = (g` k) -> (F` y) = (F` (g` k)))
17	14, 15, 16, 7	oprabval2 4957	. . . . . . 7 |- (((k + 1) e. NN /\ (g` k) e. A) -> ((k + 1){<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))} (g` k)) = (F` (g` k)))
18	12, 13, 17	syl11anc 524	. . . . . 6 |- ((g:NN-->A /\ k e. NN) -> ((k + 1){<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))} (g` k)) = (F` (g` k)))
19	18	eleq2d 1964	. . . . 5 |- ((g:NN-->A /\ k e. NN) -> ((g` (k + 1)) e. ((k + 1){<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))} (g` k)) <-> (g` (k + 1)) e. (F` (g` k))))
20	19	ralbidva 2119	. . . 4 |- (g:NN-->A -> (A.k e. NN (g` (k + 1)) e. ((k + 1){<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))} (g` k)) <-> A.k e. NN (g` (k + 1)) e. (F` (g` k))))
21	20	pm5.32i 707	. . 3 |- ((g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1){<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))} (g` k))) <-> (g:NN-->A /\ A.k e. NN (g` (k + 1)) e. (F` (g` k))))
22	21	exbii 1398	. 2 |- (E.g(g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1){<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))} (g` k))) <-> E.g(g:NN-->A /\ A.k e. NN (g` (k + 1)) e. (F` (g` k))))
23	10, 22	sylib 215	1 |- ((A =/= (/) /\ F:A-->(~PA \ {(/)})) -> E.g(g:NN-->A /\ A.k e. NN (g` (k + 1)) e. (F` (g` k))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  _Vcvv 2292   \ cdif 2590  (/)c0 2875  ~Pcpw 3032  {csn 3044   X. cxp 3984  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  1c1 6387   + caddc 6389  NNcn 6449

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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