Theorem dfac11 37632

Description: The right-hand side of this theorem (compare with ac4 9297), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 8497, despite not mentioning the cumulative hierarchy in any way as most consequences of regularity do.

This is definition (MC) of [Schechter] p. 141. EDITORIAL: the proof is not original with me of course but I lost my reference sometime after writing it.

A multiple choice function allows any total order to be extended to a choice function, which in turn defines a well-ordering. Since a well ordering on a set defines a simple ordering of the power set, this allows the trivial well-ordering of the empty set to be transfinitely bootstrapped up the cumulative hierarchy to any desired level. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Stefan O'Rear, 1-Jun-2015.)

Assertion

Ref	Expression
dfac11	|- (CHOICE <-> A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) )

Distinct variable group:   x, z, f

Proof of Theorem dfac11

Dummy variables a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfac3 8944	. . 3 |- (CHOICE <-> A. a E. c A. d e. a ( d =/= (/) -> ( c ` d ) e. d ) )
2		raleq 3138	. . . . . 6 |- ( a = x -> ( A. d e. a ( d =/= (/) -> ( c ` d ) e. d ) <-> A. d e. x ( d =/= (/) -> ( c ` d ) e. d ) ) )
3	2	exbidv 1850	. . . . 5 |- ( a = x -> ( E. c A. d e. a ( d =/= (/) -> ( c ` d ) e. d ) <-> E. c A. d e. x ( d =/= (/) -> ( c ` d ) e. d ) ) )
4	3	cbvalv 2273	. . . 4 |- ( A. a E. c A. d e. a ( d =/= (/) -> ( c ` d ) e. d ) <-> A. x E. c A. d e. x ( d =/= (/) -> ( c ` d ) e. d ) )
5		neeq1 2856	. . . . . . . . . 10 |- ( d = z -> ( d =/= (/) <-> z =/= (/) ) )
6		fveq2 6191	. . . . . . . . . . 11 |- ( d = z -> ( c ` d ) = ( c ` z ) )
7		id 22	. . . . . . . . . . 11 |- ( d = z -> d = z )
8	6, 7	eleq12d 2695	. . . . . . . . . 10 |- ( d = z -> ( ( c ` d ) e. d <-> ( c ` z ) e. z ) )
9	5, 8	imbi12d 334	. . . . . . . . 9 |- ( d = z -> ( ( d =/= (/) -> ( c ` d ) e. d ) <-> ( z =/= (/) -> ( c ` z ) e. z ) ) )
10	9	cbvralv 3171	. . . . . . . 8 |- ( A. d e. x ( d =/= (/) -> ( c ` d ) e. d ) <-> A. z e. x ( z =/= (/) -> ( c ` z ) e. z ) )
11		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( b = z -> ( c ` b ) = ( c ` z ) )
12	11	sneqd 4189	. . . . . . . . . . . . . 14 |- ( b = z -> { ( c ` b ) } = { ( c ` z ) } )
13		eqid 2622	. . . . . . . . . . . . . 14 |- ( b e. x |-> { ( c ` b ) } ) = ( b e. x |-> { ( c ` b ) } )
14		snex 4908	. . . . . . . . . . . . . 14 |- { ( c ` z ) } e. _V
15	12, 13, 14	fvmpt 6282	. . . . . . . . . . . . 13 |- ( z e. x -> ( ( b e. x |-> { ( c ` b ) } ) ` z ) = { ( c ` z ) } )
16	15	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( z e. x /\ z =/= (/) /\ ( c ` z ) e. z ) -> ( ( b e. x |-> { ( c ` b ) } ) ` z ) = { ( c ` z ) } )
17		simp3 1063	. . . . . . . . . . . . . . . 16 |- ( ( z e. x /\ z =/= (/) /\ ( c ` z ) e. z ) -> ( c ` z ) e. z )
18	17	snssd 4340	. . . . . . . . . . . . . . 15 |- ( ( z e. x /\ z =/= (/) /\ ( c ` z ) e. z ) -> { ( c ` z ) } C_ z )
19	14	elpw 4164	. . . . . . . . . . . . . . 15 |- ( { ( c ` z ) } e. ~P z <-> { ( c ` z ) } C_ z )
20	18, 19	sylibr 224	. . . . . . . . . . . . . 14 |- ( ( z e. x /\ z =/= (/) /\ ( c ` z ) e. z ) -> { ( c ` z ) } e. ~P z )
21		snfi 8038	. . . . . . . . . . . . . . 15 |- { ( c ` z ) } e. Fin
22	21	a1i 11	. . . . . . . . . . . . . 14 |- ( ( z e. x /\ z =/= (/) /\ ( c ` z ) e. z ) -> { ( c ` z ) } e. Fin )
23	20, 22	elind 3798	. . . . . . . . . . . . 13 |- ( ( z e. x /\ z =/= (/) /\ ( c ` z ) e. z ) -> { ( c ` z ) } e. ( ~P z i^i Fin ) )
24		fvex 6201	. . . . . . . . . . . . . . 15 |- ( c ` z ) e. _V
25	24	snnz 4309	. . . . . . . . . . . . . 14 |- { ( c ` z ) } =/= (/)
26	25	a1i 11	. . . . . . . . . . . . 13 |- ( ( z e. x /\ z =/= (/) /\ ( c ` z ) e. z ) -> { ( c ` z ) } =/= (/) )
27		eldifsn 4317	. . . . . . . . . . . . 13 |- ( { ( c ` z ) } e. ( ( ~P z i^i Fin ) \ { (/) } ) <-> ( { ( c ` z ) } e. ( ~P z i^i Fin ) /\ { ( c ` z ) } =/= (/) ) )
28	23, 26, 27	sylanbrc 698	. . . . . . . . . . . 12 |- ( ( z e. x /\ z =/= (/) /\ ( c ` z ) e. z ) -> { ( c ` z ) } e. ( ( ~P z i^i Fin ) \ { (/) } ) )
29	16, 28	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( z e. x /\ z =/= (/) /\ ( c ` z ) e. z ) -> ( ( b e. x |-> { ( c ` b ) } ) ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) )
30	29	3exp 1264	. . . . . . . . . 10 |- ( z e. x -> ( z =/= (/) -> ( ( c ` z ) e. z -> ( ( b e. x |-> { ( c ` b ) } ) ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) ) )
31	30	a2d 29	. . . . . . . . 9 |- ( z e. x -> ( ( z =/= (/) -> ( c ` z ) e. z ) -> ( z =/= (/) -> ( ( b e. x |-> { ( c ` b ) } ) ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) ) )
32	31	ralimia 2950	. . . . . . . 8 |- ( A. z e. x ( z =/= (/) -> ( c ` z ) e. z ) -> A. z e. x ( z =/= (/) -> ( ( b e. x |-> { ( c ` b ) } ) ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) )
33	10, 32	sylbi 207	. . . . . . 7 |- ( A. d e. x ( d =/= (/) -> ( c ` d ) e. d ) -> A. z e. x ( z =/= (/) -> ( ( b e. x |-> { ( c ` b ) } ) ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) )
34		vex 3203	. . . . . . . . 9 |- x e. _V
35	34	mptex 6486	. . . . . . . 8 |- ( b e. x |-> { ( c ` b ) } ) e. _V
36		fveq1 6190	. . . . . . . . . . 11 |- ( f = ( b e. x |-> { ( c ` b ) } ) -> ( f ` z ) = ( ( b e. x |-> { ( c ` b ) } ) ` z ) )
37	36	eleq1d 2686	. . . . . . . . . 10 |- ( f = ( b e. x |-> { ( c ` b ) } ) -> ( ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) <-> ( ( b e. x |-> { ( c ` b ) } ) ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) )
38	37	imbi2d 330	. . . . . . . . 9 |- ( f = ( b e. x |-> { ( c ` b ) } ) -> ( ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) <-> ( z =/= (/) -> ( ( b e. x |-> { ( c ` b ) } ) ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) ) )
39	38	ralbidv 2986	. . . . . . . 8 |- ( f = ( b e. x |-> { ( c ` b ) } ) -> ( A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) <-> A. z e. x ( z =/= (/) -> ( ( b e. x |-> { ( c ` b ) } ) ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) ) )
40	35, 39	spcev 3300	. . . . . . 7 |- ( A. z e. x ( z =/= (/) -> ( ( b e. x |-> { ( c ` b ) } ) ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) -> E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) )
41	33, 40	syl 17	. . . . . 6 |- ( A. d e. x ( d =/= (/) -> ( c ` d ) e. d ) -> E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) )
42	41	exlimiv 1858	. . . . 5 |- ( E. c A. d e. x ( d =/= (/) -> ( c ` d ) e. d ) -> E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) )
43	42	alimi 1739	. . . 4 |- ( A. x E. c A. d e. x ( d =/= (/) -> ( c ` d ) e. d ) -> A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) )
44	4, 43	sylbi 207	. . 3 |- ( A. a E. c A. d e. a ( d =/= (/) -> ( c ` d ) e. d ) -> A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) )
45	1, 44	sylbi 207	. 2 |- (CHOICE -> A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) )
46		fvex 6201	. . . . . . 7 |- ( R1 ` ( rank ` a ) ) e. _V
47	46	pwex 4848	. . . . . 6 |- ~P ( R1 ` ( rank ` a ) ) e. _V
48		raleq 3138	. . . . . . 7 |- ( x = ~P ( R1 ` ( rank ` a ) ) -> ( A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) <-> A. z e. ~P ( R1 ` ( rank ` a ) ) ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) ) )
49	48	exbidv 1850	. . . . . 6 |- ( x = ~P ( R1 ` ( rank ` a ) ) -> ( E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) <-> E. f A. z e. ~P ( R1 ` ( rank ` a ) ) ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) ) )
50	47, 49	spcv 3299	. . . . 5 |- ( A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) -> E. f A. z e. ~P ( R1 ` ( rank ` a ) ) ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) )
51		rankon 8658	. . . . . . . 8 |- ( rank ` a ) e. On
52	51	a1i 11	. . . . . . 7 |- ( A. z e. ~P ( R1 ` ( rank ` a ) ) ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) -> ( rank ` a ) e. On )
53		id 22	. . . . . . 7 |- ( A. z e. ~P ( R1 ` ( rank ` a ) ) ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) -> A. z e. ~P ( R1 ` ( rank ` a ) ) ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) )
54	52, 53	aomclem8 37631	. . . . . 6 |- ( A. z e. ~P ( R1 ` ( rank ` a ) ) ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) -> E. b b We ( R1 ` ( rank ` a ) ) )
55	54	exlimiv 1858	. . . . 5 |- ( E. f A. z e. ~P ( R1 ` ( rank ` a ) ) ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) -> E. b b We ( R1 ` ( rank ` a ) ) )
56		vex 3203	. . . . . 6 |- a e. _V
57		r1rankid 8722	. . . . . 6 |- ( a e. _V -> a C_ ( R1 ` ( rank ` a ) ) )
58		wess 5101	. . . . . . 7 |- ( a C_ ( R1 ` ( rank ` a ) ) -> ( b We ( R1 ` ( rank ` a ) ) -> b We a ) )
59	58	eximdv 1846	. . . . . 6 |- ( a C_ ( R1 ` ( rank ` a ) ) -> ( E. b b We ( R1 ` ( rank ` a ) ) -> E. b b We a ) )
60	56, 57, 59	mp2b 10	. . . . 5 |- ( E. b b We ( R1 ` ( rank ` a ) ) -> E. b b We a )
61	50, 55, 60	3syl 18	. . . 4 |- ( A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) -> E. b b We a )
62	61	alrimiv 1855	. . 3 |- ( A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) -> A. a E. b b We a )
63		dfac8 8957	. . 3 |- (CHOICE <-> A. a E. b b We a )
64	62, 63	sylibr 224	. 2 |- ( A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) -> CHOICE )
65	45, 64	impbii 199	1 |- (CHOICE <-> A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. ( ( ~P z i^i Fin ) \ { (/) } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   We wwe 5072   Oncon0 5723   ` cfv 5888   Fincfn 7955   R1cr1 8625   rankcrnk 8626  CHOICEwac 8938

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  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-sup 8348  df-r1 8627  df-rank 8628  df-card 8765  df-ac 8939

This theorem is referenced by: (None)