Theorem resdif 5658

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 5618	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> Fun ( F |` A ) )
2		difss 3480	. . . . . . 7 |- ( A \ B ) C_ A
3		fof 5617	. . . . . . . 8 |- ( ( F |` A ) : A -onto-> C -> ( F |` A ) : A --> C )
4		fdm 5560	. . . . . . . 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 3402	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> ( A \ B ) C_ dom ( F |` A ) )
7		fores 5626	. . . . . 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 656	. . . . 5 |- ( ( F |` A ) : A -onto-> C -> ( ( F |` A ) |` ( A \ B ) ) : ( A \ B ) -onto-> ( ( F |` A ) " ( A \ B ) ) )
9		resres 5120	. . . . . . . 8 |- ( ( F |` A ) |` ( A \ B ) ) = ( F |` ( A i^i ( A \ B ) ) )
10		indif 3589	. . . . . . . . 9 |- ( A i^i ( A \ B ) ) = ( A \ B )
11	10	reseq2i 5103	. . . . . . . 8 |- ( F |` ( A i^i ( A \ B ) ) ) = ( F |` ( A \ B ) )
12	9, 11	eqtri 2461	. . . . . . 7 |- ( ( F |` A ) |` ( A \ B ) ) = ( F |` ( A \ B ) )
13		foeq1 5613	. . . . . . 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 5062	. . . . . . . 8 |- ran ( ( F |` A ) |` ( A \ B ) ) = ran ( F |` ( A \ B ) )
16		df-ima 4849	. . . . . . . 8 |- ( ( F |` A ) " ( A \ B ) ) = ran ( ( F |` A ) |` ( A \ B ) )
17		df-ima 4849	. . . . . . . 8 |- ( F " ( A \ B ) ) = ran ( F |` ( A \ B ) )
18	15, 16, 17	3eqtr4i 2471	. . . . . . 7 |- ( ( F |` A ) " ( A \ B ) ) = ( F " ( A \ B ) )
19		foeq3 5615	. . . . . . 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 5483	. . . 4 |- ( Fun `' F -> Fun `' ( F |` ( A \ B ) ) )
24		dff1o3 5644	. . . . 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 475	. . 3 |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C ) -> ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( F " ( A \ B ) ) )
27	26	3adant3 1003	. 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 4849	. . . . . . 7 |- ( F " A ) = ran ( F |` A )
29		forn 5620	. . . . . . 7 |- ( ( F |` A ) : A -onto-> C -> ran ( F |` A ) = C )
30	28, 29	syl5eq 2485	. . . . . 6 |- ( ( F |` A ) : A -onto-> C -> ( F " A ) = C )
31		df-ima 4849	. . . . . . 7 |- ( F " B ) = ran ( F |` B )
32		forn 5620	. . . . . . 7 |- ( ( F |` B ) : B -onto-> D -> ran ( F |` B ) = D )
33	31, 32	syl5eq 2485	. . . . . 6 |- ( ( F |` B ) : B -onto-> D -> ( F " B ) = D )
34	30, 33	anim12i 563	. . . . 5 |- ( ( ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( ( F " A ) = C /\ ( F " B ) = D ) )
35		imadif 5490	. . . . . 6 |- ( Fun `' F -> ( F " ( A \ B ) ) = ( ( F " A ) \ ( F " B ) ) )
36		difeq12 3466	. . . . . 6 |- ( ( ( F " A ) = C /\ ( F " B ) = D ) -> ( ( F " A ) \ ( F " B ) ) = ( C \ D ) )
37	35, 36	sylan9eq 2493	. . . . 5 |- ( ( Fun `' F /\ ( ( F " A ) = C /\ ( F " B ) = D ) ) -> ( F " ( A \ B ) ) = ( C \ D ) )
38	34, 37	sylan2 471	. . . 4 |- ( ( Fun `' F /\ ( ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) ) -> ( F " ( A \ B ) ) = ( C \ D ) )
39	38	3impb 1178	. . 3 |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( F " ( A \ B ) ) = ( C \ D ) )
40		f1oeq3 5631	. . 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 960   = wceq 1364   \ cdif 3322   i^i cin 3324   C_ wss 3325   `'ccnv 4835   dom cdm 4836   ran crn 4837   |` cres 4838   "cima 4839   Fun wfun 5409   -->wf 5411   -onto->wfo 5413   -1-1-onto->wf1o 5414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422

This theorem is referenced by:  resin  5659  canthp1lem2  8816  subfacp1lem3  26984  subfacp1lem5  26986