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Theorem numacn 8872
Description: A well-orderable set has choice sequences of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
numacn |- ( A e. V -> ( X e. dom card -> X e. AC A ) )
Proof of Theorem numacn
Dummy variables f g h x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3212 . 2 |- ( A e. V -> A e. _V )
2 simpll 790 . . . . . . . 8 |- ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) -> X e. dom card )
3 elmapi 7879 . . . . . . . . . . . 12 |- ( f e. ( ( ~P X \ { (/) } ) ^m A ) -> f : A --> ( ~P X \ { (/) } ) )
43adantl 482 . . . . . . . . . . 11 |- ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) -> f : A --> ( ~P X \ { (/) } ) )
5 frn 6053 . . . . . . . . . . 11 |- ( f : A --> ( ~P X \ { (/) } ) -> ran f C_ ( ~P X \ { (/) } ) )
64, 5syl 17 . . . . . . . . . 10 |- ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) -> ran f C_ ( ~P X \ { (/) } ) )
76difss2d 3740 . . . . . . . . 9 |- ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) -> ran f C_ ~P X )
8 sspwuni 4611 . . . . . . . . 9 |- ( ran f C_ ~P X <-> U. ran f C_ X )
97, 8sylib 208 . . . . . . . 8 |- ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) -> U. ran f C_ X )
10 ssnum 8862 . . . . . . . 8 |- ( ( X e. dom card /\ U. ran f C_ X ) -> U. ran f e. dom card )
112, 9, 10syl2anc 693 . . . . . . 7 |- ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) -> U. ran f e. dom card )
12 ssdifin0 4050 . . . . . . . . 9 |- ( ran f C_ ( ~P X \ { (/) } ) -> ( ran f i^i { (/) } ) = (/) )
136, 12syl 17 . . . . . . . 8 |- ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) -> ( ran f i^i { (/) } ) = (/) )
14 disjsn 4246 . . . . . . . 8 |- ( ( ran f i^i { (/) } ) = (/) <-> -. (/) e. ran f )
1513, 14sylib 208 . . . . . . 7 |- ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) -> -. (/) e. ran f )
16 ac5num 8859 . . . . . . 7 |- ( ( U. ran f e. dom card /\ -. (/) e. ran f ) -> E. h ( h : ran f --> U. ran f /\ A. y e. ran f ( h ` y ) e. y ) )
1711, 15, 16syl2anc 693 . . . . . 6 |- ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) -> E. h ( h : ran f --> U. ran f /\ A. y e. ran f ( h ` y ) e. y ) )
18 simpllr 799 . . . . . . 7 |- ( ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) /\ ( h : ran f --> U. ran f /\ A. y e. ran f ( h ` y ) e. y ) ) -> A e. _V )
19 ffn 6045 . . . . . . . . . . 11 |- ( f : A --> ( ~P X \ { (/) } ) -> f Fn A )
204, 19syl 17 . . . . . . . . . 10 |- ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) -> f Fn A )
21 fveq2 6191 . . . . . . . . . . . 12 |- ( y = ( f ` x ) -> ( h ` y ) = ( h ` ( f ` x ) ) )
22 id 22 . . . . . . . . . . . 12 |- ( y = ( f ` x ) -> y = ( f ` x ) )
2321, 22eleq12d 2695 . . . . . . . . . . 11 |- ( y = ( f ` x ) -> ( ( h ` y ) e. y <-> ( h ` ( f ` x ) ) e. ( f ` x ) ) )
2423ralrn 6362 . . . . . . . . . 10 |- ( f Fn A -> ( A. y e. ran f ( h ` y ) e. y <-> A. x e. A ( h ` ( f ` x ) ) e. ( f ` x ) ) )
2520, 24syl 17 . . . . . . . . 9 |- ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) -> ( A. y e. ran f ( h ` y ) e. y <-> A. x e. A ( h ` ( f ` x ) ) e. ( f ` x ) ) )
2625biimpa 501 . . . . . . . 8 |- ( ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) /\ A. y e. ran f ( h ` y ) e. y ) -> A. x e. A ( h ` ( f ` x ) ) e. ( f ` x ) )
2726adantrl 752 . . . . . . 7 |- ( ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) /\ ( h : ran f --> U. ran f /\ A. y e. ran f ( h ` y ) e. y ) ) -> A. x e. A ( h ` ( f ` x ) ) e. ( f ` x ) )
28 acnlem 8871 . . . . . . 7 |- ( ( A e. _V /\ A. x e. A ( h ` ( f ` x ) ) e. ( f ` x ) ) -> E. g A. x e. A ( g ` x ) e. ( f ` x ) )
2918, 27, 28syl2anc 693 . . . . . 6 |- ( ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) /\ ( h : ran f --> U. ran f /\ A. y e. ran f ( h ` y ) e. y ) ) -> E. g A. x e. A ( g ` x ) e. ( f ` x ) )
3017, 29exlimddv 1863 . . . . 5 |- ( ( ( X e. dom card /\ A e. _V ) /\ f e. ( ( ~P X \ { (/) } ) ^m A ) ) -> E. g A. x e. A ( g ` x ) e. ( f ` x ) )
3130ralrimiva 2966 . . . 4 |- ( ( X e. dom card /\ A e. _V ) -> A. f e. ( ( ~P X \ { (/) } ) ^m A ) E. g A. x e. A ( g ` x ) e. ( f ` x ) )
32 isacn 8867 . . . 4 |- ( ( X e. dom card /\ A e. _V ) -> ( X e. AC A <-> A. f e. ( ( ~P X \ { (/) } ) ^m A ) E. g A. x e. A ( g ` x ) e. ( f ` x ) ) )
3331, 32mpbird 247 . . 3 |- ( ( X e. dom card /\ A e. _V ) -> X e. AC A )
3433expcom 451 . 2 |- ( A e. _V -> ( X e. dom card -> X e. AC A ) )
351, 34syl 17 1 |- ( A e. V -> ( X e. dom card -> X e. AC A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   cardccrd 8761  AC wacn 8764
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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
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-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-int 4476  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-se 5074  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-pred 5680  df-ord 5726  df-on 5727  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-card 8765  df-acn 8768
This theorem is referenced by:  acnnum  8875  fodomnum  8880  acacni  8962  dfac13  8964  iundom  9364  iunctb  9396
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