Theorem resdif 5566

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 5521	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> Fun ( F |` A ) )
2		difss 3307	. . . . . . 7 |- ( A \ B ) C_ A
3		fof 5520	. . . . . . . 8 |- ( ( F |` A ) : A -onto-> C -> ( F |` A ) : A --> C )
4		fdm 5451	. . . . . . . 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 3252	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> ( A \ B ) C_ dom ( F |` A ) )
7		fores 5530	. . . . . 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 4990	. . . . . . . 8 |- ( ( F |` A ) |` ( A \ B ) ) = ( F |` ( A i^i ( A \ B ) ) )
10		indif 3424	. . . . . . . . 9 |- ( A i^i ( A \ B ) ) = ( A \ B )
11	10	reseq2i 4975	. . . . . . . 8 |- ( F |` ( A i^i ( A \ B ) ) ) = ( F |` ( A \ B ) )
12	9, 11	eqtri 2228	. . . . . . 7 |- ( ( F |` A ) |` ( A \ B ) ) = ( F |` ( A \ B ) )
13		foeq1 5516	. . . . . . 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 4925	. . . . . . . 8 |- ran ( ( F |` A ) |` ( A \ B ) ) = ran ( F |` ( A \ B ) )
16		df-ima 4706	. . . . . . . 8 |- ( ( F |` A ) " ( A \ B ) ) = ran ( ( F |` A ) |` ( A \ B ) )
17		df-ima 4706	. . . . . . . 8 |- ( F " ( A \ B ) ) = ran ( F |` ( A \ B ) )
18	15, 16, 17	3eqtr4i 2238	. . . . . . 7 |- ( ( F |` A ) " ( A \ B ) ) = ( F " ( A \ B ) )
19		foeq3 5518	. . . . . . 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 5365	. . . 4 |- ( Fun `' F -> Fun `' ( F |` ( A \ B ) ) )
24		dff1o3 5550	. . . . 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 1020	. 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 4706	. . . . . . 7 |- ( F " A ) = ran ( F |` A )
29		forn 5523	. . . . . . 7 |- ( ( F |` A ) : A -onto-> C -> ran ( F |` A ) = C )
30	28, 29	eqtrid 2252	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> ( F " A ) = C )
31		df-ima 4706	. . . . . . 7 |- ( F " B ) = ran ( F |` B )
32		forn 5523	. . . . . . 7 |- ( ( F |` B ) : B -onto-> D -> ran ( F |` B ) = D )
33	31, 32	eqtrid 2252	. . . . . 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 5373	. . . . . 6 |- ( Fun `' F -> ( F " ( A \ B ) ) = ( ( F " A ) \ ( F " B ) ) )
36		difeq12 3294	. . . . . 6 |- ( ( ( F " A ) = C /\ ( F " B ) = D ) -> ( ( F " A ) \ ( F " B ) ) = ( C \ D ) )
37	35, 36	sylan9eq 2260	. . . . 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 1202	. . 3 |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( F " ( A \ B ) ) = ( C \ D ) )
40		f1oeq3 5534	. . 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 981   = wceq 1373   \ cdif 3171   i^i cin 3173   C_ wss 3174   `'ccnv 4692   dom cdm 4693   ran crn 4694   |` cres 4695   "cima 4696   Fun wfun 5284   -->wf 5286   -onto->wfo 5288   -1-1-onto->wf1o 5289

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297

This theorem is referenced by:  dif1en  7002