Theorem acdcALT 7496

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-->(P~A \ {(/)})) -> 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 7490	. . 3 |- ((A =/= (/) /\ {<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))}:(NN X. A)-->(P~A \ {(/)})) -> 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 3814	. . . . . . 7 |- ((F:A-->(P~A \ {(/)}) /\ y e. A) -> (F` y) e. (P~A \ {(/)}))
4	3	ex 373	. . . . . 6 |- (F:A-->(P~A \ {(/)}) -> (y e. A -> (F` y) e. (P~A \ {(/)})))
5	4	adantld 390	. . . . 5 |- (F:A-->(P~A \ {(/)}) -> ((x e. NN /\ y e. A) -> (F` y) e. (P~A \ {(/)})))
6	5	r19.21aivv 1720	. . . 4 |- (F:A-->(P~A \ {(/)}) -> A.x e. NN A.y e. A (F` y) e. (P~A \ {(/)}))
7		eqid 1475	. . . . 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 4119	. . . 4 |- (A.x e. NN A.y e. A (F` y) e. (P~A \ {(/)}) <-> {<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))}:(NN X. A)-->(P~A \ {(/)}))
9	6, 8	sylib 198	. . 3 |- (F:A-->(P~A \ {(/)}) -> {<.<.x, y>., z>. | ((x e. NN /\ y e. A) /\ z = (F` y))}:(NN X. A)-->(P~A \ {(/)}))
10	2, 9	sylan2 451	. 2 |- ((A =/= (/) /\ F:A-->(P~A \ {(/)})) -> 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		fvex 3732	. . . . . . . 8 |- (F` (g` k)) e. V
12		eqidd 1476	. . . . . . . 8 |- (x = (k + 1) -> (F` y) = (F` y))
13		fveq2 3724	. . . . . . . 8 |- (y = (g` k) -> (F` y) = (F` (g` k)))
14	11, 12, 13, 7	oprabval2 4028	. . . . . . 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)))
15		peano2nn 5935	. . . . . . . 8 |- (k e. NN -> (k + 1) e. NN)
16	15	adantl 388	. . . . . . 7 |- ((g:NN-->A /\ k e. NN) -> (k + 1) e. NN)
17		ffvelrn 3814	. . . . . . 7 |- ((g:NN-->A /\ k e. NN) -> (g` k) e. A)
18	14, 16, 17	sylanc 471	. . . . . 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 1541	. . . . 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 1659	. . . 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 645	. . 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 1051	. 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 198	1 |- ((A =/= (/) /\ F:A-->(P~A \ {(/)})) -> E.g(g:NN-->A /\ A.k e. NN (g` (k + 1)) e. (F` (g` k))))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980   =/= wne 1585  A.wral 1645  Vcvv 1811   \ cdif 2044  (/)c0 2280  P~cpw 2401  {csn 2409   X. cxp 3168  -->wf 3178  ` cfv 3182  (class class class)co 3963  {copab2 3964  1c1 5235   + caddc 5237  NNcn 5296

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-iso 3199  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-n 5925  df-n0 6100  df-z 6136  df-seq1 6308

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