Theorem axdc2 9271

Description: An apparent strengthening of ax-dc 9268 (but derived from it) which shows that there is a denumerable sequence g for any function that maps elements of a set A to nonempty subsets of A such that g ( x + 1 ) e. F ( g ( x ) ) for all x e. om. The finitistic version of this can be proven by induction, but the infinite version requires this new axiom. (Contributed by Mario Carneiro, 25-Jan-2013.)

Hypothesis

Ref	Expression
axdc2.1	|- A e. _V

Assertion

Ref	Expression
axdc2	|- ( ( A =/= (/) /\ F : A --> ( ~P A \ { (/) } ) ) -> E. g ( g : om --> A /\ A. k e. om ( g ` suc k ) e. ( F ` ( g ` k ) ) ) )

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

Proof of Theorem axdc2

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

Step	Hyp	Ref	Expression
1		axdc2.1	. 2 |- A e. _V
2		eleq1 2689	. . . . 5 |- ( s = x -> ( s e. A <-> x e. A ) )
3	2	adantr 481	. . . 4 |- ( ( s = x /\ t = y ) -> ( s e. A <-> x e. A ) )
4		fveq2 6191	. . . . . 6 |- ( s = x -> ( F ` s ) = ( F ` x ) )
5	4	eleq2d 2687	. . . . 5 |- ( s = x -> ( t e. ( F ` s ) <-> t e. ( F ` x ) ) )
6		eleq1 2689	. . . . 5 |- ( t = y -> ( t e. ( F ` x ) <-> y e. ( F ` x ) ) )
7	5, 6	sylan9bb 736	. . . 4 |- ( ( s = x /\ t = y ) -> ( t e. ( F ` s ) <-> y e. ( F ` x ) ) )
8	3, 7	anbi12d 747	. . 3 |- ( ( s = x /\ t = y ) -> ( ( s e. A /\ t e. ( F ` s ) ) <-> ( x e. A /\ y e. ( F ` x ) ) ) )
9	8	cbvopabv 4722	. 2 |- { <. s , t >. | ( s e. A /\ t e. ( F ` s ) ) } = { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) }
10		fveq2 6191	. . 3 |- ( n = x -> ( h ` n ) = ( h ` x ) )
11	10	cbvmptv 4750	. 2 |- ( n e. om |-> ( h ` n ) ) = ( x e. om |-> ( h ` x ) )
12	1, 9, 11	axdc2lem 9270	1 |- ( ( A =/= (/) /\ F : A --> ( ~P A \ { (/) } ) ) -> E. g ( g : om --> A /\ A. k e. om ( g ` suc k ) e. ( F ` ( g ` k ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   {copab 4712   |-> cmpt 4729   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-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-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-fv 5896  df-om 7066  df-1o 7560

This theorem is referenced by:  axdc3lem4  9275