Theorem fipreima 8272

Description: Given a finite subset A of the range of a function, there exists a finite subset of the domain whose image is A. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
fipreima	|- ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) -> E. c e. ( ~P B i^i Fin ) ( F " c ) = A )

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

Proof of Theorem fipreima

Dummy variables f x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1063	. . 3 |- ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) -> A e. Fin )
2		dfss3 3592	. . . . . 6 |- ( A C_ ran F <-> A. x e. A x e. ran F )
3		fvelrnb 6243	. . . . . . 7 |- ( F Fn B -> ( x e. ran F <-> E. y e. B ( F ` y ) = x ) )
4	3	ralbidv 2986	. . . . . 6 |- ( F Fn B -> ( A. x e. A x e. ran F <-> A. x e. A E. y e. B ( F ` y ) = x ) )
5	2, 4	syl5bb 272	. . . . 5 |- ( F Fn B -> ( A C_ ran F <-> A. x e. A E. y e. B ( F ` y ) = x ) )
6	5	biimpa 501	. . . 4 |- ( ( F Fn B /\ A C_ ran F ) -> A. x e. A E. y e. B ( F ` y ) = x )
7	6	3adant3 1081	. . 3 |- ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) -> A. x e. A E. y e. B ( F ` y ) = x )
8		fveq2 6191	. . . . 5 |- ( y = ( f ` x ) -> ( F ` y ) = ( F ` ( f ` x ) ) )
9	8	eqeq1d 2624	. . . 4 |- ( y = ( f ` x ) -> ( ( F ` y ) = x <-> ( F ` ( f ` x ) ) = x ) )
10	9	ac6sfi 8204	. . 3 |- ( ( A e. Fin /\ A. x e. A E. y e. B ( F ` y ) = x ) -> E. f ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) )
11	1, 7, 10	syl2anc 693	. 2 |- ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) -> E. f ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) )
12		fimass 6081	. . . . . 6 |- ( f : A --> B -> ( f " A ) C_ B )
13		vex 3203	. . . . . . . 8 |- f e. _V
14	13	imaex 7104	. . . . . . 7 |- ( f " A ) e. _V
15	14	elpw 4164	. . . . . 6 |- ( ( f " A ) e. ~P B <-> ( f " A ) C_ B )
16	12, 15	sylibr 224	. . . . 5 |- ( f : A --> B -> ( f " A ) e. ~P B )
17	16	ad2antrl 764	. . . 4 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> ( f " A ) e. ~P B )
18		ffun 6048	. . . . . 6 |- ( f : A --> B -> Fun f )
19	18	ad2antrl 764	. . . . 5 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> Fun f )
20		simpl3 1066	. . . . 5 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> A e. Fin )
21		imafi 8259	. . . . 5 |- ( ( Fun f /\ A e. Fin ) -> ( f " A ) e. Fin )
22	19, 20, 21	syl2anc 693	. . . 4 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> ( f " A ) e. Fin )
23	17, 22	elind 3798	. . 3 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> ( f " A ) e. ( ~P B i^i Fin ) )
24		fvco3 6275	. . . . . . . . . . 11 |- ( ( f : A --> B /\ x e. A ) -> ( ( F o. f ) ` x ) = ( F ` ( f ` x ) ) )
25		fvresi 6439	. . . . . . . . . . . 12 |- ( x e. A -> ( ( _I |` A ) ` x ) = x )
26	25	adantl 482	. . . . . . . . . . 11 |- ( ( f : A --> B /\ x e. A ) -> ( ( _I |` A ) ` x ) = x )
27	24, 26	eqeq12d 2637	. . . . . . . . . 10 |- ( ( f : A --> B /\ x e. A ) -> ( ( ( F o. f ) ` x ) = ( ( _I |` A ) ` x ) <-> ( F ` ( f ` x ) ) = x ) )
28	27	ralbidva 2985	. . . . . . . . 9 |- ( f : A --> B -> ( A. x e. A ( ( F o. f ) ` x ) = ( ( _I |` A ) ` x ) <-> A. x e. A ( F ` ( f ` x ) ) = x ) )
29	28	biimprd 238	. . . . . . . 8 |- ( f : A --> B -> ( A. x e. A ( F ` ( f ` x ) ) = x -> A. x e. A ( ( F o. f ) ` x ) = ( ( _I |` A ) ` x ) ) )
30	29	adantl 482	. . . . . . 7 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ f : A --> B ) -> ( A. x e. A ( F ` ( f ` x ) ) = x -> A. x e. A ( ( F o. f ) ` x ) = ( ( _I |` A ) ` x ) ) )
31	30	impr 649	. . . . . 6 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> A. x e. A ( ( F o. f ) ` x ) = ( ( _I |` A ) ` x ) )
32		simpl1 1064	. . . . . . . 8 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> F Fn B )
33		ffn 6045	. . . . . . . . 9 |- ( f : A --> B -> f Fn A )
34	33	ad2antrl 764	. . . . . . . 8 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> f Fn A )
35		frn 6053	. . . . . . . . 9 |- ( f : A --> B -> ran f C_ B )
36	35	ad2antrl 764	. . . . . . . 8 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> ran f C_ B )
37		fnco 5999	. . . . . . . 8 |- ( ( F Fn B /\ f Fn A /\ ran f C_ B ) -> ( F o. f ) Fn A )
38	32, 34, 36, 37	syl3anc 1326	. . . . . . 7 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> ( F o. f ) Fn A )
39		fnresi 6008	. . . . . . 7 |- ( _I |` A ) Fn A
40		eqfnfv 6311	. . . . . . 7 |- ( ( ( F o. f ) Fn A /\ ( _I |` A ) Fn A ) -> ( ( F o. f ) = ( _I |` A ) <-> A. x e. A ( ( F o. f ) ` x ) = ( ( _I |` A ) ` x ) ) )
41	38, 39, 40	sylancl 694	. . . . . 6 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> ( ( F o. f ) = ( _I |` A ) <-> A. x e. A ( ( F o. f ) ` x ) = ( ( _I |` A ) ` x ) ) )
42	31, 41	mpbird 247	. . . . 5 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> ( F o. f ) = ( _I |` A ) )
43	42	imaeq1d 5465	. . . 4 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> ( ( F o. f ) " A ) = ( ( _I |` A ) " A ) )
44		imaco 5640	. . . 4 |- ( ( F o. f ) " A ) = ( F " ( f " A ) )
45		ssid 3624	. . . . 5 |- A C_ A
46		resiima 5480	. . . . 5 |- ( A C_ A -> ( ( _I |` A ) " A ) = A )
47	45, 46	ax-mp 5	. . . 4 |- ( ( _I |` A ) " A ) = A
48	43, 44, 47	3eqtr3g 2679	. . 3 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> ( F " ( f " A ) ) = A )
49		imaeq2 5462	. . . . 5 |- ( c = ( f " A ) -> ( F " c ) = ( F " ( f " A ) ) )
50	49	eqeq1d 2624	. . . 4 |- ( c = ( f " A ) -> ( ( F " c ) = A <-> ( F " ( f " A ) ) = A ) )
51	50	rspcev 3309	. . 3 |- ( ( ( f " A ) e. ( ~P B i^i Fin ) /\ ( F " ( f " A ) ) = A ) -> E. c e. ( ~P B i^i Fin ) ( F " c ) = A )
52	23, 48, 51	syl2anc 693	. 2 |- ( ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) /\ ( f : A --> B /\ A. x e. A ( F ` ( f ` x ) ) = x ) ) -> E. c e. ( ~P B i^i Fin ) ( F " c ) = A )
53	11, 52	exlimddv 1863	1 |- ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) -> E. c e. ( ~P B i^i Fin ) ( F " c ) = A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   _I cid 5023   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888   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-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  df-fin 7959

This theorem is referenced by:  fodomfi2  8883  cmpfi  21211  elrfirn  37258  lmhmfgsplit  37656  hbtlem6  37699