Theorem f1opwfi 7636

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 2443	. 2 |- ( b e. ( ~P A i^i Fin ) |-> ( F " b ) ) = ( b e. ( ~P A i^i Fin ) |-> ( F " b ) )
2		imassrn 5201	. . . . . 6 |- ( F " b ) C_ ran F
3		f1ofo 5669	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -onto-> B )
4		forn 5644	. . . . . . 7 |- ( F : A -onto-> B -> ran F = B )
5	3, 4	syl 16	. . . . . 6 |- ( F : A -1-1-onto-> B -> ran F = B )
6	2, 5	syl5sseq 3425	. . . . 5 |- ( F : A -1-1-onto-> B -> ( F " b ) C_ B )
7	6	adantr 465	. . . 4 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F " b ) C_ B )
8		inss2 3592	. . . . . . 7 |- ( ~P A i^i Fin ) C_ Fin
9		simpr 461	. . . . . . 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 3375	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> b e. Fin )
11		f1ofun 5664	. . . . . . . 8 |- ( F : A -1-1-onto-> B -> Fun F )
12	11	adantr 465	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> Fun F )
13		inss1 3591	. . . . . . . . . . 11 |- ( ~P A i^i Fin ) C_ ~P A
14	13	sseli 3373	. . . . . . . . . 10 |- ( b e. ( ~P A i^i Fin ) -> b e. ~P A )
15		elpwi 3890	. . . . . . . . . 10 |- ( b e. ~P A -> b C_ A )
16	14, 15	syl 16	. . . . . . . . 9 |- ( b e. ( ~P A i^i Fin ) -> b C_ A )
17	16	adantl 466	. . . . . . . 8 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> b C_ A )
18		f1odm 5666	. . . . . . . . 9 |- ( F : A -1-1-onto-> B -> dom F = A )
19	18	adantr 465	. . . . . . . 8 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> dom F = A )
20	17, 19	sseqtr4d 3414	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> b C_ dom F )
21		fores 5650	. . . . . . 7 |- ( ( Fun F /\ b C_ dom F ) -> ( F |` b ) : b -onto-> ( F " b ) )
22	12, 20, 21	syl2anc 661	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F |` b ) : b -onto-> ( F " b ) )
23		fofi 7618	. . . . . 6 |- ( ( b e. Fin /\ ( F |` b ) : b -onto-> ( F " b ) ) -> ( F " b ) e. Fin )
24	10, 22, 23	syl2anc 661	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F " b ) e. Fin )
25		elpwg 3889	. . . . 5 |- ( ( F " b ) e. Fin -> ( ( F " b ) e. ~P B <-> ( F " b ) C_ B ) )
26	24, 25	syl 16	. . . 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 232	. . 3 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F " b ) e. ~P B )
28	27, 24	elind 3561	. 2 |- ( ( F : A -1-1-onto-> B /\ b e. ( ~P A i^i Fin ) ) -> ( F " b ) e. ( ~P B i^i Fin ) )
29		imassrn 5201	. . . . . 6 |- ( `' F " a ) C_ ran `' F
30		dfdm4 5053	. . . . . . 7 |- dom F = ran `' F
31	30, 18	syl5eqr 2489	. . . . . 6 |- ( F : A -1-1-onto-> B -> ran `' F = A )
32	29, 31	syl5sseq 3425	. . . . 5 |- ( F : A -1-1-onto-> B -> ( `' F " a ) C_ A )
33	32	adantr 465	. . . 4 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( `' F " a ) C_ A )
34		inss2 3592	. . . . . . 7 |- ( ~P B i^i Fin ) C_ Fin
35		simpr 461	. . . . . . 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 3375	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> a e. Fin )
37		dff1o3 5668	. . . . . . . . 9 |- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) )
38	37	simprbi 464	. . . . . . . 8 |- ( F : A -1-1-onto-> B -> Fun `' F )
39	38	adantr 465	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> Fun `' F )
40		inss1 3591	. . . . . . . . . . 11 |- ( ~P B i^i Fin ) C_ ~P B
41	40	sseli 3373	. . . . . . . . . 10 |- ( a e. ( ~P B i^i Fin ) -> a e. ~P B )
42	41	adantl 466	. . . . . . . . 9 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> a e. ~P B )
43		elpwi 3890	. . . . . . . . 9 |- ( a e. ~P B -> a C_ B )
44	42, 43	syl 16	. . . . . . . 8 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> a C_ B )
45		f1ocnv 5674	. . . . . . . . . 10 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
46	45	adantr 465	. . . . . . . . 9 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> `' F : B -1-1-onto-> A )
47		f1odm 5666	. . . . . . . . 9 |- ( `' F : B -1-1-onto-> A -> dom `' F = B )
48	46, 47	syl 16	. . . . . . . 8 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> dom `' F = B )
49	44, 48	sseqtr4d 3414	. . . . . . 7 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> a C_ dom `' F )
50		fores 5650	. . . . . . 7 |- ( ( Fun `' F /\ a C_ dom `' F ) -> ( `' F |` a ) : a -onto-> ( `' F " a ) )
51	39, 49, 50	syl2anc 661	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( `' F |` a ) : a -onto-> ( `' F " a ) )
52		fofi 7618	. . . . . 6 |- ( ( a e. Fin /\ ( `' F |` a ) : a -onto-> ( `' F " a ) ) -> ( `' F " a ) e. Fin )
53	36, 51, 52	syl2anc 661	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( `' F " a ) e. Fin )
54		elpwg 3889	. . . . 5 |- ( ( `' F " a ) e. Fin -> ( ( `' F " a ) e. ~P A <-> ( `' F " a ) C_ A ) )
55	53, 54	syl 16	. . . 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 232	. . 3 |- ( ( F : A -1-1-onto-> B /\ a e. ( ~P B i^i Fin ) ) -> ( `' F " a ) e. ~P A )
57	56, 53	elind 3561	. 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 566	. . 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 466	. . . . . . 7 |- ( ( b e. ~P A /\ a e. ~P B ) -> a C_ B )
60		foimacnv 5679	. . . . . . 7 |- ( ( F : A -onto-> B /\ a C_ B ) -> ( F " ( `' F " a ) ) = a )
61	3, 59, 60	syl2an 477	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( F " ( `' F " a ) ) = a )
62	61	eqcomd 2448	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> a = ( F " ( `' F " a ) ) )
63		imaeq2 5186	. . . . . 6 |- ( b = ( `' F " a ) -> ( F " b ) = ( F " ( `' F " a ) ) )
64	63	eqeq2d 2454	. . . . 5 |- ( b = ( `' F " a ) -> ( a = ( F " b ) <-> a = ( F " ( `' F " a ) ) ) )
65	62, 64	syl5ibrcom 222	. . . 4 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( b = ( `' F " a ) -> a = ( F " b ) ) )
66		f1of1 5661	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -1-1-> B )
67	15	adantr 465	. . . . . . 7 |- ( ( b e. ~P A /\ a e. ~P B ) -> b C_ A )
68		f1imacnv 5678	. . . . . . 7 |- ( ( F : A -1-1-> B /\ b C_ A ) -> ( `' F " ( F " b ) ) = b )
69	66, 67, 68	syl2an 477	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( `' F " ( F " b ) ) = b )
70	69	eqcomd 2448	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ ( b e. ~P A /\ a e. ~P B ) ) -> b = ( `' F " ( F " b ) ) )
71		imaeq2 5186	. . . . . 6 |- ( a = ( F " b ) -> ( `' F " a ) = ( `' F " ( F " b ) ) )
72	71	eqeq2d 2454	. . . . 5 |- ( a = ( F " b ) -> ( b = ( `' F " a ) <-> b = ( `' F " ( F " b ) ) ) )
73	70, 72	syl5ibrcom 222	. . . 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 191	. . 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 474	. 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 6333	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 184   /\ wa 369   = wceq 1369   e. wcel 1756   i^i cin 3348   C_ wss 3349   ~Pcpw 3881   e. cmpt 4371   `'ccnv 4860   dom cdm 4861   ran crn 4862   |` cres 4863   "cima 4864   Fun wfun 5433   -1-1->wf1 5436   -onto->wfo 5437   -1-1-onto->wf1o 5438   Fincfn 7331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-om 6498  df-1o 6941  df-er 7122  df-en 7332  df-dom 7333  df-fin 7335

This theorem is referenced by:  fictb  8435  ackbijnn  13312  tsmsf1o  19741  eulerpartgbij  26777