Theorem resdif 5464

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 5421	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> Fun ( F |` A ) )
2		difss 3253	. . . . . . 7 |- ( A \ B ) C_ A
3		fof 5420	. . . . . . . 8 |- ( ( F |` A ) : A -onto-> C -> ( F |` A ) : A --> C )
4		fdm 5353	. . . . . . . 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 3198	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> ( A \ B ) C_ dom ( F |` A ) )
7		fores 5429	. . . . . 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 409	. . . . 5 |- ( ( F |` A ) : A -onto-> C -> ( ( F |` A ) |` ( A \ B ) ) : ( A \ B ) -onto-> ( ( F |` A ) " ( A \ B ) ) )
9		resres 4903	. . . . . . . 8 |- ( ( F |` A ) |` ( A \ B ) ) = ( F |` ( A i^i ( A \ B ) ) )
10		indif 3370	. . . . . . . . 9 |- ( A i^i ( A \ B ) ) = ( A \ B )
11	10	reseq2i 4888	. . . . . . . 8 |- ( F |` ( A i^i ( A \ B ) ) ) = ( F |` ( A \ B ) )
12	9, 11	eqtri 2191	. . . . . . 7 |- ( ( F |` A ) |` ( A \ B ) ) = ( F |` ( A \ B ) )
13		foeq1 5416	. . . . . . 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 4839	. . . . . . . 8 |- ran ( ( F |` A ) |` ( A \ B ) ) = ran ( F |` ( A \ B ) )
16		df-ima 4624	. . . . . . . 8 |- ( ( F |` A ) " ( A \ B ) ) = ran ( ( F |` A ) |` ( A \ B ) )
17		df-ima 4624	. . . . . . . 8 |- ( F " ( A \ B ) ) = ran ( F |` ( A \ B ) )
18	15, 16, 17	3eqtr4i 2201	. . . . . . 7 |- ( ( F |` A ) " ( A \ B ) ) = ( F " ( A \ B ) )
19		foeq3 5418	. . . . . . 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 183	. . . . 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 121	. . . 4 |- ( ( F |` A ) : A -onto-> C -> ( F |` ( A \ B ) ) : ( A \ B ) -onto-> ( F " ( A \ B ) ) )
23		funres11 5270	. . . 4 |- ( Fun `' F -> Fun `' ( F |` ( A \ B ) ) )
24		dff1o3 5448	. . . . 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 132	. . . 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 288	. . 3 |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C ) -> ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( F " ( A \ B ) ) )
27	26	3adant3 1012	. 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 4624	. . . . . . 7 |- ( F " A ) = ran ( F |` A )
29		forn 5423	. . . . . . 7 |- ( ( F |` A ) : A -onto-> C -> ran ( F |` A ) = C )
30	28, 29	eqtrid 2215	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> ( F " A ) = C )
31		df-ima 4624	. . . . . . 7 |- ( F " B ) = ran ( F |` B )
32		forn 5423	. . . . . . 7 |- ( ( F |` B ) : B -onto-> D -> ran ( F |` B ) = D )
33	31, 32	eqtrid 2215	. . . . . 6 |- ( ( F |` B ) : B -onto-> D -> ( F " B ) = D )
34	30, 33	anim12i 336	. . . . 5 |- ( ( ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( ( F " A ) = C /\ ( F " B ) = D ) )
35		imadif 5278	. . . . . 6 |- ( Fun `' F -> ( F " ( A \ B ) ) = ( ( F " A ) \ ( F " B ) ) )
36		difeq12 3240	. . . . . 6 |- ( ( ( F " A ) = C /\ ( F " B ) = D ) -> ( ( F " A ) \ ( F " B ) ) = ( C \ D ) )
37	35, 36	sylan9eq 2223	. . . . 5 |- ( ( Fun `' F /\ ( ( F " A ) = C /\ ( F " B ) = D ) ) -> ( F " ( A \ B ) ) = ( C \ D ) )
38	34, 37	sylan2 284	. . . 4 |- ( ( Fun `' F /\ ( ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) ) -> ( F " ( A \ B ) ) = ( C \ D ) )
39	38	3impb 1194	. . 3 |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( F " ( A \ B ) ) = ( C \ D ) )
40		f1oeq3 5433	. . 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 146	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 103   <-> wb 104   /\ w3a 973   = wceq 1348   \ cdif 3118   i^i cin 3120   C_ wss 3121   `'ccnv 4610   dom cdm 4611   ran crn 4612   |` cres 4613   "cima 4614   Fun wfun 5192   -->wf 5194   -onto->wfo 5196   -1-1-onto->wf1o 5197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205

This theorem is referenced by:  dif1en  6857