Theorem resdif 5522

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 5477	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> Fun ( F |` A ) )
2		difss 3285	. . . . . . 7 |- ( A \ B ) C_ A
3		fof 5476	. . . . . . . 8 |- ( ( F |` A ) : A -onto-> C -> ( F |` A ) : A --> C )
4		fdm 5409	. . . . . . . 8 |- ( ( F |` A ) : A --> C -> dom ( F |` A ) = A )
5	3, 4	syl 14	. . . . . . 7 |- ( ( F |` A ) : A -onto-> C -> dom ( F |` A ) = A )
6	2, 5	sseqtrrid 3230	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> ( A \ B ) C_ dom ( F |` A ) )
7		fores 5486	. . . . . 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 411	. . . . 5 |- ( ( F |` A ) : A -onto-> C -> ( ( F |` A ) |` ( A \ B ) ) : ( A \ B ) -onto-> ( ( F |` A ) " ( A \ B ) ) )
9		resres 4954	. . . . . . . 8 |- ( ( F |` A ) |` ( A \ B ) ) = ( F |` ( A i^i ( A \ B ) ) )
10		indif 3402	. . . . . . . . 9 |- ( A i^i ( A \ B ) ) = ( A \ B )
11	10	reseq2i 4939	. . . . . . . 8 |- ( F |` ( A i^i ( A \ B ) ) ) = ( F |` ( A \ B ) )
12	9, 11	eqtri 2214	. . . . . . 7 |- ( ( F |` A ) |` ( A \ B ) ) = ( F |` ( A \ B ) )
13		foeq1 5472	. . . . . . 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 4890	. . . . . . . 8 |- ran ( ( F |` A ) |` ( A \ B ) ) = ran ( F |` ( A \ B ) )
16		df-ima 4672	. . . . . . . 8 |- ( ( F |` A ) " ( A \ B ) ) = ran ( ( F |` A ) |` ( A \ B ) )
17		df-ima 4672	. . . . . . . 8 |- ( F " ( A \ B ) ) = ran ( F |` ( A \ B ) )
18	15, 16, 17	3eqtr4i 2224	. . . . . . 7 |- ( ( F |` A ) " ( A \ B ) ) = ( F " ( A \ B ) )
19		foeq3 5474	. . . . . . 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 184	. . . . 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 122	. . . 4 |- ( ( F |` A ) : A -onto-> C -> ( F |` ( A \ B ) ) : ( A \ B ) -onto-> ( F " ( A \ B ) ) )
23		funres11 5326	. . . 4 |- ( Fun `' F -> Fun `' ( F |` ( A \ B ) ) )
24		dff1o3 5506	. . . . 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 133	. . . 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 290	. . 3 |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C ) -> ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( F " ( A \ B ) ) )
27	26	3adant3 1019	. 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 4672	. . . . . . 7 |- ( F " A ) = ran ( F |` A )
29		forn 5479	. . . . . . 7 |- ( ( F |` A ) : A -onto-> C -> ran ( F |` A ) = C )
30	28, 29	eqtrid 2238	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> ( F " A ) = C )
31		df-ima 4672	. . . . . . 7 |- ( F " B ) = ran ( F |` B )
32		forn 5479	. . . . . . 7 |- ( ( F |` B ) : B -onto-> D -> ran ( F |` B ) = D )
33	31, 32	eqtrid 2238	. . . . . 6 |- ( ( F |` B ) : B -onto-> D -> ( F " B ) = D )
34	30, 33	anim12i 338	. . . . 5 |- ( ( ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( ( F " A ) = C /\ ( F " B ) = D ) )
35		imadif 5334	. . . . . 6 |- ( Fun `' F -> ( F " ( A \ B ) ) = ( ( F " A ) \ ( F " B ) ) )
36		difeq12 3272	. . . . . 6 |- ( ( ( F " A ) = C /\ ( F " B ) = D ) -> ( ( F " A ) \ ( F " B ) ) = ( C \ D ) )
37	35, 36	sylan9eq 2246	. . . . 5 |- ( ( Fun `' F /\ ( ( F " A ) = C /\ ( F " B ) = D ) ) -> ( F " ( A \ B ) ) = ( C \ D ) )
38	34, 37	sylan2 286	. . . 4 |- ( ( Fun `' F /\ ( ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) ) -> ( F " ( A \ B ) ) = ( C \ D ) )
39	38	3impb 1201	. . 3 |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( F " ( A \ B ) ) = ( C \ D ) )
40		f1oeq3 5490	. . 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 14	. 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 147	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   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   \ cdif 3150   i^i cin 3152   C_ wss 3153   `'ccnv 4658   dom cdm 4659   ran crn 4660   |` cres 4661   "cima 4662   Fun wfun 5248   -->wf 5250   -onto->wfo 5252   -1-1-onto->wf1o 5253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261

This theorem is referenced by:  dif1en  6935