Theorem acdcALT 7497

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 7491	. . 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 3820	. . . . . . 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 392	. . . . 5 |- (F:A-->(P~A \ {(/)}) -> ((x e. NN /\ y e. A) -> (F` y) e. (P~A \ {(/)})))
6	5	r19.21aivv 1723	. . . 4 |- (F:A-->(P~A \ {(/)}) -> A.x e. NN A.y e. A (F` y) e. (P~A \ {(/)}))
7		eqid 1478	. . . . 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 4125	. . . 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 453	. 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 3738	. . . . . . . 8 |- (F` (g` k)) e. V
12		eqidd 1479	. . . . . . . 8 |- (x = (k + 1) -> (F` y) = (F` y))
13		fveq2 3730	. . . . . . . 8 |- (y = (g` k) -> (F` y) = (F` (g` k)))
14	11, 12, 13, 7	oprabval2 4034	. . . . . . 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 5937	. . . . . . . 8 |- (k e. NN -> (k + 1) e. NN)
16	15	adantl 390	. . . . . . 7 |- ((g:NN-->A /\ k e. NN) -> (k + 1) e. NN)
17		ffvelrn 3820	. . . . . . 7 |- ((g:NN-->A /\ k e. NN) -> (g` k) e. A)
18	14, 16, 17	sylanc 473	. . . . . 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 1544	. . . . 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 1662	. . . 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 647	. . 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 1053	. 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 958   e. wcel 960  E.wex 982   =/= wne 1588  A.wral 1648  Vcvv 1814   \ cdif 2047  (/)c0 2283  P~cpw 2405  {csn 2413   X. cxp 3174  -->wf 3184  ` cfv 3188  (class class class)co 3969  {copab2 3970  1c1 5247   + caddc 5249  NNcn 5308

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-iso 3205  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-n 5927  df-n0 6102  df-z 6138  df-seq1 6309

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