Theorem fin1a2lem6 9227

Description: Lemma for fin1a2 9237. Establish that om can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)

Hypotheses

Ref	Expression
fin1a2lem.b	|- E = ( x e. om |-> ( 2o .o x ) )
fin1a2lem.aa	|- S = ( x e. On |-> suc x )

Assertion

Ref	Expression
fin1a2lem6	|- ( S |` ran E ) : ran E -1-1-onto-> ( om \ ran E )

Proof of Theorem fin1a2lem6

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

Step	Hyp	Ref	Expression
1		fin1a2lem.aa	. . . 4 |- S = ( x e. On |-> suc x )
2	1	fin1a2lem2 9223	. . 3 |- S : On -1-1-> On
3		fin1a2lem.b	. . . . 5 |- E = ( x e. om |-> ( 2o .o x ) )
4	3	fin1a2lem4 9225	. . . 4 |- E : om -1-1-> om
5		f1f 6101	. . . 4 |- ( E : om -1-1-> om -> E : om --> om )
6		frn 6053	. . . . 5 |- ( E : om --> om -> ran E C_ om )
7		omsson 7069	. . . . 5 |- om C_ On
8	6, 7	syl6ss 3615	. . . 4 |- ( E : om --> om -> ran E C_ On )
9	4, 5, 8	mp2b 10	. . 3 |- ran E C_ On
10		f1ores 6151	. . 3 |- ( ( S : On -1-1-> On /\ ran E C_ On ) -> ( S |` ran E ) : ran E -1-1-onto-> ( S " ran E ) )
11	2, 9, 10	mp2an 708	. 2 |- ( S |` ran E ) : ran E -1-1-onto-> ( S " ran E )
12	9	sseli 3599	. . . . . . . . 9 |- ( b e. ran E -> b e. On )
13	1	fin1a2lem1 9222	. . . . . . . . 9 |- ( b e. On -> ( S ` b ) = suc b )
14	12, 13	syl 17	. . . . . . . 8 |- ( b e. ran E -> ( S ` b ) = suc b )
15	14	eqeq1d 2624	. . . . . . 7 |- ( b e. ran E -> ( ( S ` b ) = a <-> suc b = a ) )
16	15	rexbiia 3040	. . . . . 6 |- ( E. b e. ran E ( S ` b ) = a <-> E. b e. ran E suc b = a )
17	4, 5, 6	mp2b 10	. . . . . . . . . . . 12 |- ran E C_ om
18	17	sseli 3599	. . . . . . . . . . 11 |- ( b e. ran E -> b e. om )
19		peano2 7086	. . . . . . . . . . 11 |- ( b e. om -> suc b e. om )
20	18, 19	syl 17	. . . . . . . . . 10 |- ( b e. ran E -> suc b e. om )
21	3	fin1a2lem5 9226	. . . . . . . . . . . 12 |- ( b e. om -> ( b e. ran E <-> -. suc b e. ran E ) )
22	21	biimpd 219	. . . . . . . . . . 11 |- ( b e. om -> ( b e. ran E -> -. suc b e. ran E ) )
23	18, 22	mpcom 38	. . . . . . . . . 10 |- ( b e. ran E -> -. suc b e. ran E )
24	20, 23	jca 554	. . . . . . . . 9 |- ( b e. ran E -> ( suc b e. om /\ -. suc b e. ran E ) )
25		eleq1 2689	. . . . . . . . . 10 |- ( suc b = a -> ( suc b e. om <-> a e. om ) )
26		eleq1 2689	. . . . . . . . . . 11 |- ( suc b = a -> ( suc b e. ran E <-> a e. ran E ) )
27	26	notbid 308	. . . . . . . . . 10 |- ( suc b = a -> ( -. suc b e. ran E <-> -. a e. ran E ) )
28	25, 27	anbi12d 747	. . . . . . . . 9 |- ( suc b = a -> ( ( suc b e. om /\ -. suc b e. ran E ) <-> ( a e. om /\ -. a e. ran E ) ) )
29	24, 28	syl5ibcom 235	. . . . . . . 8 |- ( b e. ran E -> ( suc b = a -> ( a e. om /\ -. a e. ran E ) ) )
30	29	rexlimiv 3027	. . . . . . 7 |- ( E. b e. ran E suc b = a -> ( a e. om /\ -. a e. ran E ) )
31		peano1 7085	. . . . . . . . . . . . . 14 |- (/) e. om
32	3	fin1a2lem3 9224	. . . . . . . . . . . . . 14 |- ( (/) e. om -> ( E ` (/) ) = ( 2o .o (/) ) )
33	31, 32	ax-mp 5	. . . . . . . . . . . . 13 |- ( E ` (/) ) = ( 2o .o (/) )
34		om0x 7599	. . . . . . . . . . . . 13 |- ( 2o .o (/) ) = (/)
35	33, 34	eqtri 2644	. . . . . . . . . . . 12 |- ( E ` (/) ) = (/)
36		f1fun 6103	. . . . . . . . . . . . . 14 |- ( E : om -1-1-> om -> Fun E )
37	4, 36	ax-mp 5	. . . . . . . . . . . . 13 |- Fun E
38		f1dm 6105	. . . . . . . . . . . . . . 15 |- ( E : om -1-1-> om -> dom E = om )
39	4, 38	ax-mp 5	. . . . . . . . . . . . . 14 |- dom E = om
40	31, 39	eleqtrri 2700	. . . . . . . . . . . . 13 |- (/) e. dom E
41		fvelrn 6352	. . . . . . . . . . . . 13 |- ( ( Fun E /\ (/) e. dom E ) -> ( E ` (/) ) e. ran E )
42	37, 40, 41	mp2an 708	. . . . . . . . . . . 12 |- ( E ` (/) ) e. ran E
43	35, 42	eqeltrri 2698	. . . . . . . . . . 11 |- (/) e. ran E
44		eleq1 2689	. . . . . . . . . . 11 |- ( a = (/) -> ( a e. ran E <-> (/) e. ran E ) )
45	43, 44	mpbiri 248	. . . . . . . . . 10 |- ( a = (/) -> a e. ran E )
46	45	necon3bi 2820	. . . . . . . . 9 |- ( -. a e. ran E -> a =/= (/) )
47		nnsuc 7082	. . . . . . . . 9 |- ( ( a e. om /\ a =/= (/) ) -> E. b e. om a = suc b )
48	46, 47	sylan2 491	. . . . . . . 8 |- ( ( a e. om /\ -. a e. ran E ) -> E. b e. om a = suc b )
49		eleq1 2689	. . . . . . . . . . . . . . . 16 |- ( a = suc b -> ( a e. om <-> suc b e. om ) )
50		eleq1 2689	. . . . . . . . . . . . . . . . 17 |- ( a = suc b -> ( a e. ran E <-> suc b e. ran E ) )
51	50	notbid 308	. . . . . . . . . . . . . . . 16 |- ( a = suc b -> ( -. a e. ran E <-> -. suc b e. ran E ) )
52	49, 51	anbi12d 747	. . . . . . . . . . . . . . 15 |- ( a = suc b -> ( ( a e. om /\ -. a e. ran E ) <-> ( suc b e. om /\ -. suc b e. ran E ) ) )
53	52	anbi1d 741	. . . . . . . . . . . . . 14 |- ( a = suc b -> ( ( ( a e. om /\ -. a e. ran E ) /\ b e. om ) <-> ( ( suc b e. om /\ -. suc b e. ran E ) /\ b e. om ) ) )
54		simplr 792	. . . . . . . . . . . . . . 15 |- ( ( ( suc b e. om /\ -. suc b e. ran E ) /\ b e. om ) -> -. suc b e. ran E )
55	21	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( suc b e. om /\ -. suc b e. ran E ) /\ b e. om ) -> ( b e. ran E <-> -. suc b e. ran E ) )
56	54, 55	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( ( suc b e. om /\ -. suc b e. ran E ) /\ b e. om ) -> b e. ran E )
57	53, 56	syl6bi 243	. . . . . . . . . . . . 13 |- ( a = suc b -> ( ( ( a e. om /\ -. a e. ran E ) /\ b e. om ) -> b e. ran E ) )
58	57	com12 32	. . . . . . . . . . . 12 |- ( ( ( a e. om /\ -. a e. ran E ) /\ b e. om ) -> ( a = suc b -> b e. ran E ) )
59	58	impr 649	. . . . . . . . . . 11 |- ( ( ( a e. om /\ -. a e. ran E ) /\ ( b e. om /\ a = suc b ) ) -> b e. ran E )
60		simprr 796	. . . . . . . . . . . 12 |- ( ( ( a e. om /\ -. a e. ran E ) /\ ( b e. om /\ a = suc b ) ) -> a = suc b )
61	60	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ( a e. om /\ -. a e. ran E ) /\ ( b e. om /\ a = suc b ) ) -> suc b = a )
62	59, 61	jca 554	. . . . . . . . . 10 |- ( ( ( a e. om /\ -. a e. ran E ) /\ ( b e. om /\ a = suc b ) ) -> ( b e. ran E /\ suc b = a ) )
63	62	ex 450	. . . . . . . . 9 |- ( ( a e. om /\ -. a e. ran E ) -> ( ( b e. om /\ a = suc b ) -> ( b e. ran E /\ suc b = a ) ) )
64	63	reximdv2 3014	. . . . . . . 8 |- ( ( a e. om /\ -. a e. ran E ) -> ( E. b e. om a = suc b -> E. b e. ran E suc b = a ) )
65	48, 64	mpd 15	. . . . . . 7 |- ( ( a e. om /\ -. a e. ran E ) -> E. b e. ran E suc b = a )
66	30, 65	impbii 199	. . . . . 6 |- ( E. b e. ran E suc b = a <-> ( a e. om /\ -. a e. ran E ) )
67	16, 66	bitri 264	. . . . 5 |- ( E. b e. ran E ( S ` b ) = a <-> ( a e. om /\ -. a e. ran E ) )
68		f1fn 6102	. . . . . . 7 |- ( S : On -1-1-> On -> S Fn On )
69	2, 68	ax-mp 5	. . . . . 6 |- S Fn On
70		fvelimab 6253	. . . . . 6 |- ( ( S Fn On /\ ran E C_ On ) -> ( a e. ( S " ran E ) <-> E. b e. ran E ( S ` b ) = a ) )
71	69, 9, 70	mp2an 708	. . . . 5 |- ( a e. ( S " ran E ) <-> E. b e. ran E ( S ` b ) = a )
72		eldif 3584	. . . . 5 |- ( a e. ( om \ ran E ) <-> ( a e. om /\ -. a e. ran E ) )
73	67, 71, 72	3bitr4i 292	. . . 4 |- ( a e. ( S " ran E ) <-> a e. ( om \ ran E ) )
74	73	eqriv 2619	. . 3 |- ( S " ran E ) = ( om \ ran E )
75		f1oeq3 6129	. . 3 |- ( ( S " ran E ) = ( om \ ran E ) -> ( ( S |` ran E ) : ran E -1-1-onto-> ( S " ran E ) <-> ( S |` ran E ) : ran E -1-1-onto-> ( om \ ran E ) ) )
76	74, 75	ax-mp 5	. 2 |- ( ( S |` ran E ) : ran E -1-1-onto-> ( S " ran E ) <-> ( S |` ran E ) : ran E -1-1-onto-> ( om \ ran E ) )
77	11, 76	mpbi 220	1 |- ( S |` ran E ) : ran E -1-1-onto-> ( om \ ran E )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Oncon0 5723   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   omcom 7065   2oc2o 7554   .o comu 7558

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-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-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-omul 7565

This theorem is referenced by:  fin1a2lem7  9228