Theorem ackbij1b 9061

Description: The Ackermann bijection, part 1b: the bijection from ackbij1 9060 restricts naturally to the powers of particular naturals. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Hypothesis

Ref	Expression
ackbij.f	|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )

Assertion

Ref	Expression
ackbij1b	|- ( A e. om -> ( F " ~P A ) = ( card ` ~P A ) )

Distinct variable groups:   x, F, y   x, A, y

Proof of Theorem ackbij1b

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ackbij2lem1 9041	. . . . 5 |- ( A e. om -> ~P A C_ ( ~P om i^i Fin ) )
2		pwexg 4850	. . . . 5 |- ( A e. om -> ~P A e. _V )
3		ackbij.f	. . . . . . 7 |- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )
4	3	ackbij1lem17 9058	. . . . . 6 |- F : ( ~P om i^i Fin ) -1-1-> om
5		f1imaeng 8016	. . . . . 6 |- ( ( F : ( ~P om i^i Fin ) -1-1-> om /\ ~P A C_ ( ~P om i^i Fin ) /\ ~P A e. _V ) -> ( F " ~P A ) ~~ ~P A )
6	4, 5	mp3an1 1411	. . . . 5 |- ( ( ~P A C_ ( ~P om i^i Fin ) /\ ~P A e. _V ) -> ( F " ~P A ) ~~ ~P A )
7	1, 2, 6	syl2anc 693	. . . 4 |- ( A e. om -> ( F " ~P A ) ~~ ~P A )
8		nnfi 8153	. . . . . 6 |- ( A e. om -> A e. Fin )
9		pwfi 8261	. . . . . 6 |- ( A e. Fin <-> ~P A e. Fin )
10	8, 9	sylib 208	. . . . 5 |- ( A e. om -> ~P A e. Fin )
11		ficardid 8788	. . . . 5 |- ( ~P A e. Fin -> ( card ` ~P A ) ~~ ~P A )
12		ensym 8005	. . . . 5 |- ( ( card ` ~P A ) ~~ ~P A -> ~P A ~~ ( card ` ~P A ) )
13	10, 11, 12	3syl 18	. . . 4 |- ( A e. om -> ~P A ~~ ( card ` ~P A ) )
14		entr 8008	. . . 4 |- ( ( ( F " ~P A ) ~~ ~P A /\ ~P A ~~ ( card ` ~P A ) ) -> ( F " ~P A ) ~~ ( card ` ~P A ) )
15	7, 13, 14	syl2anc 693	. . 3 |- ( A e. om -> ( F " ~P A ) ~~ ( card ` ~P A ) )
16		onfin2 8152	. . . . . . 7 |- om = ( On i^i Fin )
17		inss2 3834	. . . . . . 7 |- ( On i^i Fin ) C_ Fin
18	16, 17	eqsstri 3635	. . . . . 6 |- om C_ Fin
19		ficardom 8787	. . . . . . 7 |- ( ~P A e. Fin -> ( card ` ~P A ) e. om )
20	10, 19	syl 17	. . . . . 6 |- ( A e. om -> ( card ` ~P A ) e. om )
21	18, 20	sseldi 3601	. . . . 5 |- ( A e. om -> ( card ` ~P A ) e. Fin )
22		php3 8146	. . . . . 6 |- ( ( ( card ` ~P A ) e. Fin /\ ( F " ~P A ) C. ( card ` ~P A ) ) -> ( F " ~P A ) ~< ( card ` ~P A ) )
23	22	ex 450	. . . . 5 |- ( ( card ` ~P A ) e. Fin -> ( ( F " ~P A ) C. ( card ` ~P A ) -> ( F " ~P A ) ~< ( card ` ~P A ) ) )
24	21, 23	syl 17	. . . 4 |- ( A e. om -> ( ( F " ~P A ) C. ( card ` ~P A ) -> ( F " ~P A ) ~< ( card ` ~P A ) ) )
25		sdomnen 7984	. . . 4 |- ( ( F " ~P A ) ~< ( card ` ~P A ) -> -. ( F " ~P A ) ~~ ( card ` ~P A ) )
26	24, 25	syl6 35	. . 3 |- ( A e. om -> ( ( F " ~P A ) C. ( card ` ~P A ) -> -. ( F " ~P A ) ~~ ( card ` ~P A ) ) )
27	15, 26	mt2d 131	. 2 |- ( A e. om -> -. ( F " ~P A ) C. ( card ` ~P A ) )
28		fvex 6201	. . . . . 6 |- ( F ` a ) e. _V
29		ackbij1lem3 9044	. . . . . . . . 9 |- ( A e. om -> A e. ( ~P om i^i Fin ) )
30		elpwi 4168	. . . . . . . . 9 |- ( a e. ~P A -> a C_ A )
31	3	ackbij1lem12 9053	. . . . . . . . 9 |- ( ( A e. ( ~P om i^i Fin ) /\ a C_ A ) -> ( F ` a ) C_ ( F ` A ) )
32	29, 30, 31	syl2an 494	. . . . . . . 8 |- ( ( A e. om /\ a e. ~P A ) -> ( F ` a ) C_ ( F ` A ) )
33	3	ackbij1lem10 9051	. . . . . . . . . . 11 |- F : ( ~P om i^i Fin ) --> om
34		peano1 7085	. . . . . . . . . . 11 |- (/) e. om
35	33, 34	f0cli 6370	. . . . . . . . . 10 |- ( F ` a ) e. om
36		nnord 7073	. . . . . . . . . 10 |- ( ( F ` a ) e. om -> Ord ( F ` a ) )
37	35, 36	ax-mp 5	. . . . . . . . 9 |- Ord ( F ` a )
38	33, 34	f0cli 6370	. . . . . . . . . 10 |- ( F ` A ) e. om
39		nnord 7073	. . . . . . . . . 10 |- ( ( F ` A ) e. om -> Ord ( F ` A ) )
40	38, 39	ax-mp 5	. . . . . . . . 9 |- Ord ( F ` A )
41		ordsucsssuc 7023	. . . . . . . . 9 |- ( ( Ord ( F ` a ) /\ Ord ( F ` A ) ) -> ( ( F ` a ) C_ ( F ` A ) <-> suc ( F ` a ) C_ suc ( F ` A ) ) )
42	37, 40, 41	mp2an 708	. . . . . . . 8 |- ( ( F ` a ) C_ ( F ` A ) <-> suc ( F ` a ) C_ suc ( F ` A ) )
43	32, 42	sylib 208	. . . . . . 7 |- ( ( A e. om /\ a e. ~P A ) -> suc ( F ` a ) C_ suc ( F ` A ) )
44	3	ackbij1lem14 9055	. . . . . . . . 9 |- ( A e. om -> ( F ` { A } ) = suc ( F ` A ) )
45	3	ackbij1lem8 9049	. . . . . . . . 9 |- ( A e. om -> ( F ` { A } ) = ( card ` ~P A ) )
46	44, 45	eqtr3d 2658	. . . . . . . 8 |- ( A e. om -> suc ( F ` A ) = ( card ` ~P A ) )
47	46	adantr 481	. . . . . . 7 |- ( ( A e. om /\ a e. ~P A ) -> suc ( F ` A ) = ( card ` ~P A ) )
48	43, 47	sseqtrd 3641	. . . . . 6 |- ( ( A e. om /\ a e. ~P A ) -> suc ( F ` a ) C_ ( card ` ~P A ) )
49		sucssel 5819	. . . . . 6 |- ( ( F ` a ) e. _V -> ( suc ( F ` a ) C_ ( card ` ~P A ) -> ( F ` a ) e. ( card ` ~P A ) ) )
50	28, 48, 49	mpsyl 68	. . . . 5 |- ( ( A e. om /\ a e. ~P A ) -> ( F ` a ) e. ( card ` ~P A ) )
51	50	ralrimiva 2966	. . . 4 |- ( A e. om -> A. a e. ~P A ( F ` a ) e. ( card ` ~P A ) )
52		f1fun 6103	. . . . . 6 |- ( F : ( ~P om i^i Fin ) -1-1-> om -> Fun F )
53	4, 52	ax-mp 5	. . . . 5 |- Fun F
54		f1dm 6105	. . . . . . 7 |- ( F : ( ~P om i^i Fin ) -1-1-> om -> dom F = ( ~P om i^i Fin ) )
55	4, 54	ax-mp 5	. . . . . 6 |- dom F = ( ~P om i^i Fin )
56	1, 55	syl6sseqr 3652	. . . . 5 |- ( A e. om -> ~P A C_ dom F )
57		funimass4 6247	. . . . 5 |- ( ( Fun F /\ ~P A C_ dom F ) -> ( ( F " ~P A ) C_ ( card ` ~P A ) <-> A. a e. ~P A ( F ` a ) e. ( card ` ~P A ) ) )
58	53, 56, 57	sylancr 695	. . . 4 |- ( A e. om -> ( ( F " ~P A ) C_ ( card ` ~P A ) <-> A. a e. ~P A ( F ` a ) e. ( card ` ~P A ) ) )
59	51, 58	mpbird 247	. . 3 |- ( A e. om -> ( F " ~P A ) C_ ( card ` ~P A ) )
60		sspss 3706	. . 3 |- ( ( F " ~P A ) C_ ( card ` ~P A ) <-> ( ( F " ~P A ) C. ( card ` ~P A ) \/ ( F " ~P A ) = ( card ` ~P A ) ) )
61	59, 60	sylib 208	. 2 |- ( A e. om -> ( ( F " ~P A ) C. ( card ` ~P A ) \/ ( F " ~P A ) = ( card ` ~P A ) ) )
62		orel1 397	. 2 |- ( -. ( F " ~P A ) C. ( card ` ~P A ) -> ( ( ( F " ~P A ) C. ( card ` ~P A ) \/ ( F " ~P A ) = ( card ` ~P A ) ) -> ( F " ~P A ) = ( card ` ~P A ) ) )
63	27, 61, 62	sylc 65	1 |- ( A e. om -> ( F " ~P A ) = ( card ` ~P A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   C. wpss 3575   ~Pcpw 4158   {csn 4177   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   "cima 5117   Ord word 5722   Oncon0 5723   suc csuc 5725   Fun wfun 5882   -1-1->wf1 5885   ` cfv 5888   omcom 7065   ~~ cen 7952   ~< csdm 7954   Fincfn 7955   cardccrd 8761

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-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-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-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990

This theorem is referenced by:  ackbij2lem2  9062