Theorem axdc3 9276

Description: Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value C. 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. (Contributed by Mario Carneiro, 27-Jan-2013.)

Hypothesis

Ref	Expression
axdc3.1	|- A e. _V

Assertion

Ref	Expression
axdc3	|- ( ( C e. A /\ F : A --> ( ~P A \ { (/) } ) ) -> E. g ( g : om --> A /\ ( g ` (/) ) = C /\ A. k e. om ( g ` suc k ) e. ( F ` ( g ` k ) ) ) )

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

Proof of Theorem axdc3

Dummy variables n s t x y j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axdc3.1	. 2 |- A e. _V
2		feq1 6026	. . . . 5 |- ( t = s -> ( t : suc n --> A <-> s : suc n --> A ) )
3		fveq1 6190	. . . . . 6 |- ( t = s -> ( t ` (/) ) = ( s ` (/) ) )
4	3	eqeq1d 2624	. . . . 5 |- ( t = s -> ( ( t ` (/) ) = C <-> ( s ` (/) ) = C ) )
5		fveq1 6190	. . . . . . . 8 |- ( t = s -> ( t ` suc j ) = ( s ` suc j ) )
6		fveq1 6190	. . . . . . . . 9 |- ( t = s -> ( t ` j ) = ( s ` j ) )
7	6	fveq2d 6195	. . . . . . . 8 |- ( t = s -> ( F ` ( t ` j ) ) = ( F ` ( s ` j ) ) )
8	5, 7	eleq12d 2695	. . . . . . 7 |- ( t = s -> ( ( t ` suc j ) e. ( F ` ( t ` j ) ) <-> ( s ` suc j ) e. ( F ` ( s ` j ) ) ) )
9	8	ralbidv 2986	. . . . . 6 |- ( t = s -> ( A. j e. n ( t ` suc j ) e. ( F ` ( t ` j ) ) <-> A. j e. n ( s ` suc j ) e. ( F ` ( s ` j ) ) ) )
10		suceq 5790	. . . . . . . . 9 |- ( j = k -> suc j = suc k )
11	10	fveq2d 6195	. . . . . . . 8 |- ( j = k -> ( s ` suc j ) = ( s ` suc k ) )
12		fveq2 6191	. . . . . . . . 9 |- ( j = k -> ( s ` j ) = ( s ` k ) )
13	12	fveq2d 6195	. . . . . . . 8 |- ( j = k -> ( F ` ( s ` j ) ) = ( F ` ( s ` k ) ) )
14	11, 13	eleq12d 2695	. . . . . . 7 |- ( j = k -> ( ( s ` suc j ) e. ( F ` ( s ` j ) ) <-> ( s ` suc k ) e. ( F ` ( s ` k ) ) ) )
15	14	cbvralv 3171	. . . . . 6 |- ( A. j e. n ( s ` suc j ) e. ( F ` ( s ` j ) ) <-> A. k e. n ( s ` suc k ) e. ( F ` ( s ` k ) ) )
16	9, 15	syl6bb 276	. . . . 5 |- ( t = s -> ( A. j e. n ( t ` suc j ) e. ( F ` ( t ` j ) ) <-> A. k e. n ( s ` suc k ) e. ( F ` ( s ` k ) ) ) )
17	2, 4, 16	3anbi123d 1399	. . . 4 |- ( t = s -> ( ( t : suc n --> A /\ ( t ` (/) ) = C /\ A. j e. n ( t ` suc j ) e. ( F ` ( t ` j ) ) ) <-> ( s : suc n --> A /\ ( s ` (/) ) = C /\ A. k e. n ( s ` suc k ) e. ( F ` ( s ` k ) ) ) ) )
18	17	rexbidv 3052	. . 3 |- ( t = s -> ( E. n e. om ( t : suc n --> A /\ ( t ` (/) ) = C /\ A. j e. n ( t ` suc j ) e. ( F ` ( t ` j ) ) ) <-> E. n e. om ( s : suc n --> A /\ ( s ` (/) ) = C /\ A. k e. n ( s ` suc k ) e. ( F ` ( s ` k ) ) ) ) )
19	18	cbvabv 2747	. 2 |- { t | E. n e. om ( t : suc n --> A /\ ( t ` (/) ) = C /\ A. j e. n ( t ` suc j ) e. ( F ` ( t ` j ) ) ) } = { s | E. n e. om ( s : suc n --> A /\ ( s ` (/) ) = C /\ A. k e. n ( s ` suc k ) e. ( F ` ( s ` k ) ) ) }
20		eqid 2622	. 2 |- ( x e. { t | E. n e. om ( t : suc n --> A /\ ( t ` (/) ) = C /\ A. j e. n ( t ` suc j ) e. ( F ` ( t ` j ) ) ) } |-> { y e. { t | E. n e. om ( t : suc n --> A /\ ( t ` (/) ) = C /\ A. j e. n ( t ` suc j ) e. ( F ` ( t ` j ) ) ) } | ( dom y = suc dom x /\ ( y |` dom x ) = x ) } ) = ( x e. { t | E. n e. om ( t : suc n --> A /\ ( t ` (/) ) = C /\ A. j e. n ( t ` suc j ) e. ( F ` ( t ` j ) ) ) } |-> { y e. { t | E. n e. om ( t : suc n --> A /\ ( t ` (/) ) = C /\ A. j e. n ( t ` suc j ) e. ( F ` ( t ` j ) ) ) } | ( dom y = suc dom x /\ ( y |` dom x ) = x ) } )
21	1, 19, 20	axdc3lem4 9275	1 |- ( ( C e. A /\ F : A --> ( ~P A \ { (/) } ) ) -> E. g ( g : om --> A /\ ( g ` (/) ) = C /\ A. k e. om ( g ` suc k ) e. ( F ` ( g ` k ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   dom cdm 5114   |` cres 5116   suc csuc 5725   -->wf 5884   ` cfv 5888   omcom 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-dc 9268

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560

This theorem is referenced by:  axdc4lem  9277