Theorem infdifsn 8554

Description: Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
infdifsn	|- ( om ~<_ A -> ( A \ { B } ) ~~ A )

Proof of Theorem infdifsn

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 7966	. . . 4 |- ( om ~<_ A -> E. f f : om -1-1-> A )
2	1	adantr 481	. . 3 |- ( ( om ~<_ A /\ B e. A ) -> E. f f : om -1-1-> A )
3		reldom 7961	. . . . . . 7 |- Rel ~<_
4	3	brrelex2i 5159	. . . . . 6 |- ( om ~<_ A -> A e. _V )
5	4	ad2antrr 762	. . . . 5 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> A e. _V )
6		simplr 792	. . . . 5 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> B e. A )
7		f1f 6101	. . . . . . 7 |- ( f : om -1-1-> A -> f : om --> A )
8	7	adantl 482	. . . . . 6 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> f : om --> A )
9		peano1 7085	. . . . . 6 |- (/) e. om
10		ffvelrn 6357	. . . . . 6 |- ( ( f : om --> A /\ (/) e. om ) -> ( f ` (/) ) e. A )
11	8, 9, 10	sylancl 694	. . . . 5 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( f ` (/) ) e. A )
12		difsnen 8042	. . . . 5 |- ( ( A e. _V /\ B e. A /\ ( f ` (/) ) e. A ) -> ( A \ { B } ) ~~ ( A \ { ( f ` (/) ) } ) )
13	5, 6, 11, 12	syl3anc 1326	. . . 4 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( A \ { B } ) ~~ ( A \ { ( f ` (/) ) } ) )
14		vex 3203	. . . . . . . . . 10 |- f e. _V
15		f1f1orn 6148	. . . . . . . . . . 11 |- ( f : om -1-1-> A -> f : om -1-1-onto-> ran f )
16	15	adantl 482	. . . . . . . . . 10 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> f : om -1-1-onto-> ran f )
17		f1oen3g 7971	. . . . . . . . . 10 |- ( ( f e. _V /\ f : om -1-1-onto-> ran f ) -> om ~~ ran f )
18	14, 16, 17	sylancr 695	. . . . . . . . 9 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> om ~~ ran f )
19	18	ensymd 8007	. . . . . . . 8 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ran f ~~ om )
20	3	brrelexi 5158	. . . . . . . . . . 11 |- ( om ~<_ A -> om e. _V )
21	20	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> om e. _V )
22		limom 7080	. . . . . . . . . . 11 |- Lim om
23	22	limenpsi 8135	. . . . . . . . . 10 |- ( om e. _V -> om ~~ ( om \ { (/) } ) )
24	21, 23	syl 17	. . . . . . . . 9 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> om ~~ ( om \ { (/) } ) )
25	14	resex 5443	. . . . . . . . . . 11 |- ( f |` ( om \ { (/) } ) ) e. _V
26		simpr 477	. . . . . . . . . . . 12 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> f : om -1-1-> A )
27		difss 3737	. . . . . . . . . . . 12 |- ( om \ { (/) } ) C_ om
28		f1ores 6151	. . . . . . . . . . . 12 |- ( ( f : om -1-1-> A /\ ( om \ { (/) } ) C_ om ) -> ( f |` ( om \ { (/) } ) ) : ( om \ { (/) } ) -1-1-onto-> ( f " ( om \ { (/) } ) ) )
29	26, 27, 28	sylancl 694	. . . . . . . . . . 11 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( f |` ( om \ { (/) } ) ) : ( om \ { (/) } ) -1-1-onto-> ( f " ( om \ { (/) } ) ) )
30		f1oen3g 7971	. . . . . . . . . . 11 |- ( ( ( f |` ( om \ { (/) } ) ) e. _V /\ ( f |` ( om \ { (/) } ) ) : ( om \ { (/) } ) -1-1-onto-> ( f " ( om \ { (/) } ) ) ) -> ( om \ { (/) } ) ~~ ( f " ( om \ { (/) } ) ) )
31	25, 29, 30	sylancr 695	. . . . . . . . . 10 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( om \ { (/) } ) ~~ ( f " ( om \ { (/) } ) ) )
32		f1orn 6147	. . . . . . . . . . . . 13 |- ( f : om -1-1-onto-> ran f <-> ( f Fn om /\ Fun `' f ) )
33	32	simprbi 480	. . . . . . . . . . . 12 |- ( f : om -1-1-onto-> ran f -> Fun `' f )
34		imadif 5973	. . . . . . . . . . . 12 |- ( Fun `' f -> ( f " ( om \ { (/) } ) ) = ( ( f " om ) \ ( f " { (/) } ) ) )
35	16, 33, 34	3syl 18	. . . . . . . . . . 11 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( f " ( om \ { (/) } ) ) = ( ( f " om ) \ ( f " { (/) } ) ) )
36		f1fn 6102	. . . . . . . . . . . . . 14 |- ( f : om -1-1-> A -> f Fn om )
37	36	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> f Fn om )
38		fnima 6010	. . . . . . . . . . . . 13 |- ( f Fn om -> ( f " om ) = ran f )
39	37, 38	syl 17	. . . . . . . . . . . 12 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( f " om ) = ran f )
40		fnsnfv 6258	. . . . . . . . . . . . . 14 |- ( ( f Fn om /\ (/) e. om ) -> { ( f ` (/) ) } = ( f " { (/) } ) )
41	37, 9, 40	sylancl 694	. . . . . . . . . . . . 13 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> { ( f ` (/) ) } = ( f " { (/) } ) )
42	41	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( f " { (/) } ) = { ( f ` (/) ) } )
43	39, 42	difeq12d 3729	. . . . . . . . . . 11 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( ( f " om ) \ ( f " { (/) } ) ) = ( ran f \ { ( f ` (/) ) } ) )
44	35, 43	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( f " ( om \ { (/) } ) ) = ( ran f \ { ( f ` (/) ) } ) )
45	31, 44	breqtrd 4679	. . . . . . . . 9 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( om \ { (/) } ) ~~ ( ran f \ { ( f ` (/) ) } ) )
46		entr 8008	. . . . . . . . 9 |- ( ( om ~~ ( om \ { (/) } ) /\ ( om \ { (/) } ) ~~ ( ran f \ { ( f ` (/) ) } ) ) -> om ~~ ( ran f \ { ( f ` (/) ) } ) )
47	24, 45, 46	syl2anc 693	. . . . . . . 8 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> om ~~ ( ran f \ { ( f ` (/) ) } ) )
48		entr 8008	. . . . . . . 8 |- ( ( ran f ~~ om /\ om ~~ ( ran f \ { ( f ` (/) ) } ) ) -> ran f ~~ ( ran f \ { ( f ` (/) ) } ) )
49	19, 47, 48	syl2anc 693	. . . . . . 7 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ran f ~~ ( ran f \ { ( f ` (/) ) } ) )
50		difexg 4808	. . . . . . . 8 |- ( A e. _V -> ( A \ ran f ) e. _V )
51		enrefg 7987	. . . . . . . 8 |- ( ( A \ ran f ) e. _V -> ( A \ ran f ) ~~ ( A \ ran f ) )
52	5, 50, 51	3syl 18	. . . . . . 7 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( A \ ran f ) ~~ ( A \ ran f ) )
53		disjdif 4040	. . . . . . . 8 |- ( ran f i^i ( A \ ran f ) ) = (/)
54	53	a1i 11	. . . . . . 7 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( ran f i^i ( A \ ran f ) ) = (/) )
55		difss 3737	. . . . . . . . . 10 |- ( ran f \ { ( f ` (/) ) } ) C_ ran f
56		ssrin 3838	. . . . . . . . . 10 |- ( ( ran f \ { ( f ` (/) ) } ) C_ ran f -> ( ( ran f \ { ( f ` (/) ) } ) i^i ( A \ ran f ) ) C_ ( ran f i^i ( A \ ran f ) ) )
57	55, 56	ax-mp 5	. . . . . . . . 9 |- ( ( ran f \ { ( f ` (/) ) } ) i^i ( A \ ran f ) ) C_ ( ran f i^i ( A \ ran f ) )
58		sseq0 3975	. . . . . . . . 9 |- ( ( ( ( ran f \ { ( f ` (/) ) } ) i^i ( A \ ran f ) ) C_ ( ran f i^i ( A \ ran f ) ) /\ ( ran f i^i ( A \ ran f ) ) = (/) ) -> ( ( ran f \ { ( f ` (/) ) } ) i^i ( A \ ran f ) ) = (/) )
59	57, 53, 58	mp2an 708	. . . . . . . 8 |- ( ( ran f \ { ( f ` (/) ) } ) i^i ( A \ ran f ) ) = (/)
60	59	a1i 11	. . . . . . 7 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( ( ran f \ { ( f ` (/) ) } ) i^i ( A \ ran f ) ) = (/) )
61		unen 8040	. . . . . . 7 |- ( ( ( ran f ~~ ( ran f \ { ( f ` (/) ) } ) /\ ( A \ ran f ) ~~ ( A \ ran f ) ) /\ ( ( ran f i^i ( A \ ran f ) ) = (/) /\ ( ( ran f \ { ( f ` (/) ) } ) i^i ( A \ ran f ) ) = (/) ) ) -> ( ran f u. ( A \ ran f ) ) ~~ ( ( ran f \ { ( f ` (/) ) } ) u. ( A \ ran f ) ) )
62	49, 52, 54, 60, 61	syl22anc 1327	. . . . . 6 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( ran f u. ( A \ ran f ) ) ~~ ( ( ran f \ { ( f ` (/) ) } ) u. ( A \ ran f ) ) )
63		frn 6053	. . . . . . . 8 |- ( f : om --> A -> ran f C_ A )
64	8, 63	syl 17	. . . . . . 7 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ran f C_ A )
65		undif 4049	. . . . . . 7 |- ( ran f C_ A <-> ( ran f u. ( A \ ran f ) ) = A )
66	64, 65	sylib 208	. . . . . 6 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( ran f u. ( A \ ran f ) ) = A )
67		uncom 3757	. . . . . . 7 |- ( ( ran f \ { ( f ` (/) ) } ) u. ( A \ ran f ) ) = ( ( A \ ran f ) u. ( ran f \ { ( f ` (/) ) } ) )
68		eldifn 3733	. . . . . . . . . . 11 |- ( ( f ` (/) ) e. ( A \ ran f ) -> -. ( f ` (/) ) e. ran f )
69		fnfvelrn 6356	. . . . . . . . . . . 12 |- ( ( f Fn om /\ (/) e. om ) -> ( f ` (/) ) e. ran f )
70	37, 9, 69	sylancl 694	. . . . . . . . . . 11 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( f ` (/) ) e. ran f )
71	68, 70	nsyl3 133	. . . . . . . . . 10 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> -. ( f ` (/) ) e. ( A \ ran f ) )
72		disjsn 4246	. . . . . . . . . 10 |- ( ( ( A \ ran f ) i^i { ( f ` (/) ) } ) = (/) <-> -. ( f ` (/) ) e. ( A \ ran f ) )
73	71, 72	sylibr 224	. . . . . . . . 9 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( ( A \ ran f ) i^i { ( f ` (/) ) } ) = (/) )
74		undif4 4035	. . . . . . . . 9 |- ( ( ( A \ ran f ) i^i { ( f ` (/) ) } ) = (/) -> ( ( A \ ran f ) u. ( ran f \ { ( f ` (/) ) } ) ) = ( ( ( A \ ran f ) u. ran f ) \ { ( f ` (/) ) } ) )
75	73, 74	syl 17	. . . . . . . 8 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( ( A \ ran f ) u. ( ran f \ { ( f ` (/) ) } ) ) = ( ( ( A \ ran f ) u. ran f ) \ { ( f ` (/) ) } ) )
76		uncom 3757	. . . . . . . . . 10 |- ( ( A \ ran f ) u. ran f ) = ( ran f u. ( A \ ran f ) )
77	76, 66	syl5eq 2668	. . . . . . . . 9 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( ( A \ ran f ) u. ran f ) = A )
78	77	difeq1d 3727	. . . . . . . 8 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( ( ( A \ ran f ) u. ran f ) \ { ( f ` (/) ) } ) = ( A \ { ( f ` (/) ) } ) )
79	75, 78	eqtrd 2656	. . . . . . 7 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( ( A \ ran f ) u. ( ran f \ { ( f ` (/) ) } ) ) = ( A \ { ( f ` (/) ) } ) )
80	67, 79	syl5eq 2668	. . . . . 6 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( ( ran f \ { ( f ` (/) ) } ) u. ( A \ ran f ) ) = ( A \ { ( f ` (/) ) } ) )
81	62, 66, 80	3brtr3d 4684	. . . . 5 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> A ~~ ( A \ { ( f ` (/) ) } ) )
82	81	ensymd 8007	. . . 4 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( A \ { ( f ` (/) ) } ) ~~ A )
83		entr 8008	. . . 4 |- ( ( ( A \ { B } ) ~~ ( A \ { ( f ` (/) ) } ) /\ ( A \ { ( f ` (/) ) } ) ~~ A ) -> ( A \ { B } ) ~~ A )
84	13, 82, 83	syl2anc 693	. . 3 |- ( ( ( om ~<_ A /\ B e. A ) /\ f : om -1-1-> A ) -> ( A \ { B } ) ~~ A )
85	2, 84	exlimddv 1863	. 2 |- ( ( om ~<_ A /\ B e. A ) -> ( A \ { B } ) ~~ A )
86		difsn 4328	. . . 4 |- ( -. B e. A -> ( A \ { B } ) = A )
87	86	adantl 482	. . 3 |- ( ( om ~<_ A /\ -. B e. A ) -> ( A \ { B } ) = A )
88		enrefg 7987	. . . . 5 |- ( A e. _V -> A ~~ A )
89	4, 88	syl 17	. . . 4 |- ( om ~<_ A -> A ~~ A )
90	89	adantr 481	. . 3 |- ( ( om ~<_ A /\ -. B e. A ) -> A ~~ A )
91	87, 90	eqbrtrd 4675	. 2 |- ( ( om ~<_ A /\ -. B e. A ) -> ( A \ { B } ) ~~ A )
92	85, 91	pm2.61dan 832	1 |- ( om ~<_ A -> ( A \ { B } ) ~~ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888   omcom 7065   ~~ cen 7952   ~<_ cdom 7953

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

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-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-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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957

This theorem is referenced by:  infdiffi  8555  infcda1  9015  infpss  9039