Theorem f1opwfi 7878

Description: A one-to-one mapping induces a one-to-one mapping on finite subsets. (Contributed by Mario Carneiro, 25-Jan-2015.)

Assertion

Ref	Expression
f1opwfi	|- ( F : A -1-1-onto-> B -> ( b e. ( ~P A i^i Fin ) |-> ( F " b ) ) : ( ~P A i^i Fin ) -1-1-onto-> ( ~P B i^i Fin ) )

Distinct variable groups:   A, b   B, b   F, b

Proof of Theorem f1opwfi

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2451	. 2 |- ( b e. ( ~P A i^i Fin ) |-> ( F " b ) ) = ( b e. ( ~P A i^i Fin ) |-> ( F " b ) )
2		imassrn 5179	. . . . . 6 |- ( F " b ) C_ ran F
3		f1ofo 5821	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -onto-> B )
4		forn 5796	. . . . . . 7 |- ( F : A -onto-> B -> ran F = B )
5	3, 4	syl 17	. . . . . 6 |- ( F : A -1-1-onto-> B -> ran F = B )
6	2, 5	syl5sseq 3480	. . . . 5 |- ( F : A -1-1-onto-> B -> ( F " b ) C_ B )
7	6	adantr 467	. . . 4 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F " b ) C_ B )
8		inss2 3653	. . . . . . 7 |- ( ~P A i^i Fin ) C_ Fin
9		simpr 463	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> b e. ( ~P A i^i Fin ) )
10	8, 9	sseldi 3430	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> b e. Fin )
11		f1ofun 5816	. . . . . . . 8 |- ( F : A -1-1-onto-> B -> Fun F )
12	11	adantr 467	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> Fun F )
13		inss1 3652	. . . . . . . . . . 11 |- ( ~P A i^i Fin ) C_ ~P A
14	13	sseli 3428	. . . . . . . . . 10 |- ( b e. ( ~P A i^i Fin ) -> b e. ~P A )
15		elpwi 3960	. . . . . . . . . 10 |- ( b e. ~P A -> b C_ A )
16	14, 15	syl 17	. . . . . . . . 9 |- ( b e. ( ~P A i^i Fin ) -> b C_ A )
17	16	adantl 468	. . . . . . . 8 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> b C_ A )
18		f1odm 5818	. . . . . . . . 9 |- ( F : A -1-1-onto-> B -> dom F = A )
19	18	adantr 467	. . . . . . . 8 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> dom F = A )
20	17, 19	sseqtr4d 3469	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> b C_ dom F )
21		fores 5802	. . . . . . 7 |- ( ( Fun F /\ b C_ dom F ) -> ( F |` b ) : b -onto-> ( F " b ) )
22	12, 20, 21	syl2anc 667	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F |` b ) : b -onto-> ( F " b ) )
23		fofi 7860	. . . . . 6 |- ( ( b e. Fin /\ ( F |` b ) : b -onto-> ( F " b ) ) -> ( F " b ) e. Fin )
24	10, 22, 23	syl2anc 667	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F " b ) e. Fin )
25		elpwg 3959	. . . . 5 |- ( ( F " b ) e. Fin -> ( ( F " b ) e. ~P B <-> ( F " b ) C_ B ) )
26	24, 25	syl 17	. . . 4 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( ( F " b ) e. ~P B <-> ( F " b ) C_ B ) )
27	7, 26	mpbird 236	. . 3 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F " b ) e. ~P B )
28	27, 24	elind 3618	. 2 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F " b ) e. ( ~P B i^i Fin ) )
29		imassrn 5179	. . . . . 6 |- ( `' F " a ) C_ ran `' F
30		dfdm4 5027	. . . . . . 7 |- dom F = ran `' F
31	30, 18	syl5eqr 2499	. . . . . 6 |- ( F : A -1-1-onto-> B -> ran `' F = A )
32	29, 31	syl5sseq 3480	. . . . 5 |- ( F : A -1-1-onto-> B -> ( `' F " a ) C_ A )
33	32	adantr 467	. . . 4 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( `' F " a ) C_ A )
34		inss2 3653	. . . . . . 7 |- ( ~P B i^i Fin ) C_ Fin
35		simpr 463	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> a e. ( ~P B i^i Fin ) )
36	34, 35	sseldi 3430	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> a e. Fin )
37		dff1o3 5820	. . . . . . . . 9 |- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) )
38	37	simprbi 466	. . . . . . . 8 |- ( F : A -1-1-onto-> B -> Fun `' F )
39	38	adantr 467	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> Fun `' F )
40		inss1 3652	. . . . . . . . . . 11 |- ( ~P B i^i Fin ) C_ ~P B
41	40	sseli 3428	. . . . . . . . . 10 |- ( a e. ( ~P B i^i Fin ) -> a e. ~P B )
42	41	adantl 468	. . . . . . . . 9 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> a e. ~P B )
43		elpwi 3960	. . . . . . . . 9 |- ( a e. ~P B -> a C_ B )
44	42, 43	syl 17	. . . . . . . 8 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> a C_ B )
45		f1ocnv 5826	. . . . . . . . . 10 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
46	45	adantr 467	. . . . . . . . 9 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> `' F : B -1-1-onto-> A )
47		f1odm 5818	. . . . . . . . 9 |- ( `' F : B -1-1-onto-> A -> dom `' F = B )
48	46, 47	syl 17	. . . . . . . 8 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> dom `' F = B )
49	44, 48	sseqtr4d 3469	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> a C_ dom `' F )
50		fores 5802	. . . . . . 7 |- ( ( Fun `' F /\ a C_ dom `' F ) -> ( `' F |` a ) : a -onto-> ( `' F " a ) )
51	39, 49, 50	syl2anc 667	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( `' F |` a ) : a -onto-> ( `' F " a ) )
52		fofi 7860	. . . . . 6 |- ( ( a e. Fin /\ ( `' F |` a ) : a -onto-> ( `' F " a ) ) -> ( `' F " a ) e. Fin )
53	36, 51, 52	syl2anc 667	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( `' F " a ) e. Fin )
54		elpwg 3959	. . . . 5 |- ( ( `' F " a ) e. Fin -> ( ( `' F " a ) e. ~P A <-> ( `' F " a ) C_ A ) )
55	53, 54	syl 17	. . . 4 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( ( `' F " a ) e. ~P A <-> ( `' F " a ) C_ A ) )
56	33, 55	mpbird 236	. . 3 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( `' F " a ) e. ~P A )
57	56, 53	elind 3618	. 2 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( `' F " a ) e. ( ~P A i^i Fin ) )
58	14, 41	anim12i 570	. . 3 |- ( ( b e. ( ~P A i^i Fin ) /\ a e. ( ~P B i^i Fin ) ) -> ( b e. ~P A /\ a e. ~P B ) )
59	43	adantl 468	. . . . . . 7 |- ( ( b e. ~P A /\ a e. ~P B ) -> a C_ B )
60		foimacnv 5831	. . . . . . 7 |- ( ( F : A -onto-> B /\ a C_ B ) -> ( F " ( `' F " a ) ) = a )
61	3, 59, 60	syl2an 480	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( F " ( `' F " a ) ) = a )
62	61	eqcomd 2457	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> a = ( F " ( `' F " a ) ) )
63		imaeq2 5164	. . . . . 6 |- ( b = ( `' F " a ) -> ( F " b ) = ( F " ( `' F " a ) ) )
64	63	eqeq2d 2461	. . . . 5 |- ( b = ( `' F " a ) -> ( a = ( F " b ) <-> a = ( F " ( `' F " a ) ) ) )
65	62, 64	syl5ibrcom 226	. . . 4 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( b = ( `' F " a ) -> a = ( F " b ) ) )
66		f1of1 5813	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -1-1-> B )
67	15	adantr 467	. . . . . . 7 |- ( ( b e. ~P A /\ a e. ~P B ) -> b C_ A )
68		f1imacnv 5830	. . . . . . 7 |- ( ( F : A -1-1-> B /\ b C_ A ) -> ( `' F " ( F " b ) ) = b )
69	66, 67, 68	syl2an 480	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( `' F " ( F " b ) ) = b )
70	69	eqcomd 2457	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> b = ( `' F " ( F " b ) ) )
71		imaeq2 5164	. . . . . 6 |- ( a = ( F " b ) -> ( `' F " a ) = ( `' F " ( F " b ) ) )
72	71	eqeq2d 2461	. . . . 5 |- ( a = ( F " b ) -> ( b = ( `' F " a ) <-> b = ( `' F " ( F " b ) ) ) )
73	70, 72	syl5ibrcom 226	. . . 4 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( a = ( F " b ) -> b = ( `' F " a ) ) )
74	65, 73	impbid 194	. . 3 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( b = ( `' F " a ) <-> a = ( F " b ) ) )
75	58, 74	sylan2 477	. 2 |- ( ( F : A -1-1-onto-> B /\ ( b e. ( ~P A i^i Fin ) /\ a e. ( ~P B i^i Fin ) ) ) -> ( b = ( `' F " a ) <-> a = ( F " b ) ) )
76	1, 28, 57, 75	f1o2d 6521	1 |- ( F : A -1-1-onto-> B -> ( b e. ( ~P A i^i Fin ) |-> ( F " b ) ) : ( ~P A i^i Fin ) -1-1-onto-> ( ~P B i^i Fin ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   i^i cin 3403   C_ wss 3404   ~Pcpw 3951   |-> cmpt 4461   `'ccnv 4833   dom cdm 4834   ran crn 4835   |` cres 4836   "cima 4837   Fun wfun 5576   -1-1->wf1 5579   -onto->wfo 5580   -1-1-onto->wf1o 5581   Fincfn 7569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-1o 7182  df-er 7363  df-en 7570  df-dom 7571  df-fin 7573

This theorem is referenced by:  fictb  8675  ackbijnn  13886  tsmsf1o  21159  eulerpartgbij  29205