Theorem resdif 5834

Description: The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.)

Assertion

Ref	Expression
resdif	|- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( C \ D ) )

Proof of Theorem resdif

Step	Hyp	Ref	Expression
1		fofun 5794	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> Fun ( F |` A ) )
2		difss 3631	. . . . . . 7 |- ( A \ B ) C_ A
3		fof 5793	. . . . . . . 8 |- ( ( F |` A ) : A -onto-> C -> ( F |` A ) : A --> C )
4		fdm 5733	. . . . . . . 8 |- ( ( F |` A ) : A --> C -> dom ( F |` A ) = A )
5	3, 4	syl 16	. . . . . . 7 |- ( ( F |` A ) : A -onto-> C -> dom ( F |` A ) = A )
6	2, 5	syl5sseqr 3553	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> ( A \ B ) C_ dom ( F |` A ) )
7		fores 5802	. . . . . 6 |- ( ( Fun ( F |` A ) /\ ( A \ B ) C_ dom ( F |` A ) ) -> ( ( F |` A ) |` ( A \ B ) ) : ( A \ B ) -onto-> ( ( F |` A ) " ( A \ B ) ) )
8	1, 6, 7	syl2anc 661	. . . . 5 |- ( ( F |` A ) : A -onto-> C -> ( ( F |` A ) |` ( A \ B ) ) : ( A \ B ) -onto-> ( ( F |` A ) " ( A \ B ) ) )
9		resres 5284	. . . . . . . 8 |- ( ( F |` A ) |` ( A \ B ) ) = ( F |` ( A i^i ( A \ B ) ) )
10		indif 3740	. . . . . . . . 9 |- ( A i^i ( A \ B ) ) = ( A \ B )
11	10	reseq2i 5268	. . . . . . . 8 |- ( F |` ( A i^i ( A \ B ) ) ) = ( F |` ( A \ B ) )
12	9, 11	eqtri 2496	. . . . . . 7 |- ( ( F |` A ) |` ( A \ B ) ) = ( F |` ( A \ B ) )
13		foeq1 5789	. . . . . . 7 |- ( ( ( F |` A ) |` ( A \ B ) ) = ( F |` ( A \ B ) ) -> ( ( ( F |` A ) |` ( A \ B ) ) : ( A \ B ) -onto-> ( ( F |` A ) " ( A \ B ) ) <-> ( F |` ( A \ B ) ) : ( A \ B ) -onto-> ( ( F |` A ) " ( A \ B ) ) ) )
14	12, 13	ax-mp 5	. . . . . 6 |- ( ( ( F |` A ) |` ( A \ B ) ) : ( A \ B ) -onto-> ( ( F |` A ) " ( A \ B ) ) <-> ( F |` ( A \ B ) ) : ( A \ B ) -onto-> ( ( F |` A ) " ( A \ B ) ) )
15	12	rneqi 5227	. . . . . . . 8 |- ran ( ( F |` A ) |` ( A \ B ) ) = ran ( F |` ( A \ B ) )
16		df-ima 5012	. . . . . . . 8 |- ( ( F |` A ) " ( A \ B ) ) = ran ( ( F |` A ) |` ( A \ B ) )
17		df-ima 5012	. . . . . . . 8 |- ( F " ( A \ B ) ) = ran ( F |` ( A \ B ) )
18	15, 16, 17	3eqtr4i 2506	. . . . . . 7 |- ( ( F |` A ) " ( A \ B ) ) = ( F " ( A \ B ) )
19		foeq3 5791	. . . . . . 7 |- ( ( ( F |` A ) " ( A \ B ) ) = ( F " ( A \ B ) ) -> ( ( F |` ( A \ B ) ) : ( A \ B ) -onto-> ( ( F |` A ) " ( A \ B ) ) <-> ( F |` ( A \ B ) ) : ( A \ B ) -onto-> ( F " ( A \ B ) ) ) )
20	18, 19	ax-mp 5	. . . . . 6 |- ( ( F |` ( A \ B ) ) : ( A \ B ) -onto-> ( ( F |` A ) " ( A \ B ) ) <-> ( F |` ( A \ B ) ) : ( A \ B ) -onto-> ( F " ( A \ B ) ) )
21	14, 20	bitri 249	. . . . 5 |- ( ( ( F |` A ) |` ( A \ B ) ) : ( A \ B ) -onto-> ( ( F |` A ) " ( A \ B ) ) <-> ( F |` ( A \ B ) ) : ( A \ B ) -onto-> ( F " ( A \ B ) ) )
22	8, 21	sylib 196	. . . 4 |- ( ( F |` A ) : A -onto-> C -> ( F |` ( A \ B ) ) : ( A \ B ) -onto-> ( F " ( A \ B ) ) )
23		funres11 5654	. . . 4 |- ( Fun `' F -> Fun `' ( F |` ( A \ B ) ) )
24		dff1o3 5820	. . . . 5 |- ( ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( F " ( A \ B ) ) <-> ( ( F |` ( A \ B ) ) : ( A \ B ) -onto-> ( F " ( A \ B ) ) /\ Fun `' ( F |` ( A \ B ) ) ) )
25	24	biimpri 206	. . . 4 |- ( ( ( F |` ( A \ B ) ) : ( A \ B ) -onto-> ( F " ( A \ B ) ) /\ Fun `' ( F |` ( A \ B ) ) ) -> ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( F " ( A \ B ) ) )
26	22, 23, 25	syl2anr 478	. . 3 |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C ) -> ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( F " ( A \ B ) ) )
27	26	3adant3 1016	. 2 |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( F " ( A \ B ) ) )
28		df-ima 5012	. . . . . . 7 |- ( F " A ) = ran ( F |` A )
29		forn 5796	. . . . . . 7 |- ( ( F |` A ) : A -onto-> C -> ran ( F |` A ) = C )
30	28, 29	syl5eq 2520	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> ( F " A ) = C )
31		df-ima 5012	. . . . . . 7 |- ( F " B ) = ran ( F |` B )
32		forn 5796	. . . . . . 7 |- ( ( F |` B ) : B -onto-> D -> ran ( F |` B ) = D )
33	31, 32	syl5eq 2520	. . . . . 6 |- ( ( F |` B ) : B -onto-> D -> ( F " B ) = D )
34	30, 33	anim12i 566	. . . . 5 |- ( ( ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( ( F " A ) = C /\ ( F " B ) = D ) )
35		imadif 5661	. . . . . 6 |- ( Fun `' F -> ( F " ( A \ B ) ) = ( ( F " A ) \ ( F " B ) ) )
36		difeq12 3617	. . . . . 6 |- ( ( ( F " A ) = C /\ ( F " B ) = D ) -> ( ( F " A ) \ ( F " B ) ) = ( C \ D ) )
37	35, 36	sylan9eq 2528	. . . . 5 |- ( ( Fun `' F /\ ( ( F " A ) = C /\ ( F " B ) = D ) ) -> ( F " ( A \ B ) ) = ( C \ D ) )
38	34, 37	sylan2 474	. . . 4 |- ( ( Fun `' F /\ ( ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) ) -> ( F " ( A \ B ) ) = ( C \ D ) )
39	38	3impb 1192	. . 3 |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( F " ( A \ B ) ) = ( C \ D ) )
40		f1oeq3 5807	. . 3 |- ( ( F " ( A \ B ) ) = ( C \ D ) -> ( ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( F " ( A \ B ) ) <-> ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( C \ D ) ) )
41	39, 40	syl 16	. 2 |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( F " ( A \ B ) ) <-> ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( C \ D ) ) )
42	27, 41	mpbid 210	1 |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( C \ D ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   \ cdif 3473   i^i cin 3475   C_ wss 3476   `'ccnv 4998   dom cdm 4999   ran crn 5000   |` cres 5001   "cima 5002   Fun wfun 5580   -->wf 5582   -onto->wfo 5584   -1-1-onto->wf1o 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593

This theorem is referenced by:  resin  5835  canthp1lem2  9027  subfacp1lem3  28266  subfacp1lem5  28268