Theorem acdcALT 7446

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 7440	. . 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 3805	. . . . . . 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 1717	. . . 4 |- (F:A-->(P~A \ {(/)}) -> A.x e. NN A.y e. A (F` y) e. (P~A \ {(/)}))
7		eqid 1473	. . . . 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 4109	. . . 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 3723	. . . . . . . 8 |- (F` (g` k)) e. V
12		eqid 1473	. . . . . . . . 9 |- (F` y) = (F` y)
13	12	a1i 8	. . . . . . . 8 |- (x = (k + 1) -> (F` y) = (F` y))
14		fveq2 3715	. . . . . . . 8 |- (y = (g` k) -> (F` y) = (F` (g` k)))
15	11, 13, 14, 7	oprabval2 4019	. . . . . . 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)))
16		peano2nn 5891	. . . . . . . 8 |- (k e. NN -> (k + 1) e. NN)
17	16	adantl 388	. . . . . . 7 |- ((g:NN-->A /\ k e. NN) -> (k + 1) e. NN)
18		ffvelrn 3805	. . . . . . 7 |- ((g:NN-->A /\ k e. NN) -> (g` k) e. A)
19	15, 17, 18	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)))
20	19	eleq2d 1538	. . . . 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))))
21	20	ralbidva 1656	. . . 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))))
22	21	pm5.32i 644	. . 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))))
23	22	exbii 1049	. 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))))
24	10, 23	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 954   e. wcel 956  E.wex 978   =/= wne 1582  A.wral 1642  Vcvv 1807   \ cdif 2040  (/)c0 2276  P~cpw 2397  {csn 2405   X. cxp 3163  -->wf 3173  ` cfv 3177  (class class class)co 3954  {copab2 3955  1c1 5215   + caddc 5217  NNcn 5276

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-iso 3194  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-n 5881  df-n0 6055  df-z 6091  df-seq1 6253

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