Theorem f1opwfi 8270

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 2622	. 2 |- ( b e. ( ~P A i^i Fin ) |-> ( F " b ) ) = ( b e. ( ~P A i^i Fin ) |-> ( F " b ) )
2		imassrn 5477	. . . . . 6 |- ( F " b ) C_ ran F
3		f1ofo 6144	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -onto-> B )
4		forn 6118	. . . . . . 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 3653	. . . . 5 |- ( F : A -1-1-onto-> B -> ( F " b ) C_ B )
7	6	adantr 481	. . . 4 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F " b ) C_ B )
8		inss2 3834	. . . . . . 7 |- ( ~P A i^i Fin ) C_ Fin
9		simpr 477	. . . . . . 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 3601	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> b e. Fin )
11		f1ofun 6139	. . . . . . . 8 |- ( F : A -1-1-onto-> B -> Fun F )
12	11	adantr 481	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> Fun F )
13		inss1 3833	. . . . . . . . . . 11 |- ( ~P A i^i Fin ) C_ ~P A
14	13	sseli 3599	. . . . . . . . . 10 |- ( b e. ( ~P A i^i Fin ) -> b e. ~P A )
15		elpwi 4168	. . . . . . . . . 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 482	. . . . . . . 8 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> b C_ A )
18		f1odm 6141	. . . . . . . . 9 |- ( F : A -1-1-onto-> B -> dom F = A )
19	18	adantr 481	. . . . . . . 8 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> dom F = A )
20	17, 19	sseqtr4d 3642	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> b C_ dom F )
21		fores 6124	. . . . . . 7 |- ( ( Fun F /\ b C_ dom F ) -> ( F |` b ) : b -onto-> ( F " b ) )
22	12, 20, 21	syl2anc 693	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F |` b ) : b -onto-> ( F " b ) )
23		fofi 8252	. . . . . 6 |- ( ( b e. Fin /\ ( F |` b ) : b -onto-> ( F " b ) ) -> ( F " b ) e. Fin )
24	10, 22, 23	syl2anc 693	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F " b ) e. Fin )
25		elpwg 4166	. . . . 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 247	. . 3 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F " b ) e. ~P B )
28	27, 24	elind 3798	. 2 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F " b ) e. ( ~P B i^i Fin ) )
29		imassrn 5477	. . . . . 6 |- ( `' F " a ) C_ ran `' F
30		dfdm4 5316	. . . . . . 7 |- dom F = ran `' F
31	30, 18	syl5eqr 2670	. . . . . 6 |- ( F : A -1-1-onto-> B -> ran `' F = A )
32	29, 31	syl5sseq 3653	. . . . 5 |- ( F : A -1-1-onto-> B -> ( `' F " a ) C_ A )
33	32	adantr 481	. . . 4 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( `' F " a ) C_ A )
34		inss2 3834	. . . . . . 7 |- ( ~P B i^i Fin ) C_ Fin
35		simpr 477	. . . . . . 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 3601	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> a e. Fin )
37		dff1o3 6143	. . . . . . . . 9 |- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) )
38	37	simprbi 480	. . . . . . . 8 |- ( F : A -1-1-onto-> B -> Fun `' F )
39	38	adantr 481	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> Fun `' F )
40		inss1 3833	. . . . . . . . . . 11 |- ( ~P B i^i Fin ) C_ ~P B
41	40	sseli 3599	. . . . . . . . . 10 |- ( a e. ( ~P B i^i Fin ) -> a e. ~P B )
42	41	adantl 482	. . . . . . . . 9 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> a e. ~P B )
43		elpwi 4168	. . . . . . . . 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 6149	. . . . . . . . . 10 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
46	45	adantr 481	. . . . . . . . 9 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> `' F : B -1-1-onto-> A )
47		f1odm 6141	. . . . . . . . 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 3642	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> a C_ dom `' F )
50		fores 6124	. . . . . . 7 |- ( ( Fun `' F /\ a C_ dom `' F ) -> ( `' F |` a ) : a -onto-> ( `' F " a ) )
51	39, 49, 50	syl2anc 693	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( `' F |` a ) : a -onto-> ( `' F " a ) )
52		fofi 8252	. . . . . 6 |- ( ( a e. Fin /\ ( `' F |` a ) : a -onto-> ( `' F " a ) ) -> ( `' F " a ) e. Fin )
53	36, 51, 52	syl2anc 693	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( `' F " a ) e. Fin )
54		elpwg 4166	. . . . 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 247	. . 3 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( `' F " a ) e. ~P A )
57	56, 53	elind 3798	. 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 590	. . 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 482	. . . . . . 7 |- ( ( b e. ~P A /\ a e. ~P B ) -> a C_ B )
60		foimacnv 6154	. . . . . . 7 |- ( ( F : A -onto-> B /\ a C_ B ) -> ( F " ( `' F " a ) ) = a )
61	3, 59, 60	syl2an 494	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( F " ( `' F " a ) ) = a )
62	61	eqcomd 2628	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> a = ( F " ( `' F " a ) ) )
63		imaeq2 5462	. . . . . 6 |- ( b = ( `' F " a ) -> ( F " b ) = ( F " ( `' F " a ) ) )
64	63	eqeq2d 2632	. . . . 5 |- ( b = ( `' F " a ) -> ( a = ( F " b ) <-> a = ( F " ( `' F " a ) ) ) )
65	62, 64	syl5ibrcom 237	. . . 4 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( b = ( `' F " a ) -> a = ( F " b ) ) )
66		f1of1 6136	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -1-1-> B )
67	15	adantr 481	. . . . . . 7 |- ( ( b e. ~P A /\ a e. ~P B ) -> b C_ A )
68		f1imacnv 6153	. . . . . . 7 |- ( ( F : A -1-1-> B /\ b C_ A ) -> ( `' F " ( F " b ) ) = b )
69	66, 67, 68	syl2an 494	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( `' F " ( F " b ) ) = b )
70	69	eqcomd 2628	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> b = ( `' F " ( F " b ) ) )
71		imaeq2 5462	. . . . . 6 |- ( a = ( F " b ) -> ( `' F " a ) = ( `' F " ( F " b ) ) )
72	71	eqeq2d 2632	. . . . 5 |- ( a = ( F " b ) -> ( b = ( `' F " a ) <-> b = ( `' F " ( F " b ) ) ) )
73	70, 72	syl5ibrcom 237	. . . 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 202	. . 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 491	. 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 6887	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 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   Fincfn 7955

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  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  df-fin 7959

This theorem is referenced by:  fictb  9067  ackbijnn  14560  tsmsf1o  21948  eulerpartgbij  30434