Theorem resdif 4659

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		dff1o3 4641	. . . . . 6 |- ((F |` (A \ B)):(A \ B)-1-1-onto->(F"(A \ B)) <-> ((F |` (A \ B)):(A \ B)-onto->(F"(A \ B)) /\ Fun `'(F |` (A \ B))))
2	1	biimpri 169	. . . . 5 |- (((F |` (A \ B)):(A \ B)-onto->(F"(A \ B)) /\ Fun `'(F |` (A \ B))) -> (F |` (A \ B)):(A \ B)-1-1-onto->(F"(A \ B)))
3		fofun 4618	. . . . . . 7 |- ((F |` A):A-onto->C -> Fun (F |` A))
4		difss 2735	. . . . . . . 8 |- (A \ B) C_ A
5		fof 4617	. . . . . . . . . 10 |- ((F |` A):A-onto->C -> (F |` A):A-->C)
6		fdm 4567	. . . . . . . . . 10 |- ((F |` A):A-->C -> dom ( F |` A) = A)
7	5, 6	syl 12	. . . . . . . . 9 |- ((F |` A):A-onto->C -> dom ( F |` A) = A)
8	7	sseq2d 2645	. . . . . . . 8 |- ((F |` A):A-onto->C -> ((A \ B) C_ dom ( F |` A) <-> (A \ B) C_ A))
9	4, 8	mpbiri 211	. . . . . . 7 |- ((F |` A):A-onto->C -> (A \ B) C_ dom ( F |` A))
10		fores 4627	. . . . . . 7 |- ((Fun (F |` A) /\ (A \ B) C_ dom ( F |` A)) -> ((F |` A) |` (A \ B)):(A \ B)-onto->((F |` A)"(A \ B)))
11	3, 9, 10	syl11anc 524	. . . . . 6 |- ((F |` A):A-onto->C -> ((F |` A) |` (A \ B)):(A \ B)-onto->((F |` A)"(A \ B)))
12		resres 4228	. . . . . . . . 9 |- ((F |` A) |` (A \ B)) = (F |` (A i^i (A \ B)))
13		indif 2837	. . . . . . . . . 10 |- (A i^i (A \ B)) = (A \ B)
14		reseq2 4219	. . . . . . . . . 10 |- ((A i^i (A \ B)) = (A \ B) -> (F |` (A i^i (A \ B))) = (F |` (A \ B)))
15	13, 14	ax-mp 7	. . . . . . . . 9 |- (F |` (A i^i (A \ B))) = (F |` (A \ B))
16	12, 15	eqtri 1908	. . . . . . . 8 |- ((F |` A) |` (A \ B)) = (F |` (A \ B))
17		foeq1 4613	. . . . . . . 8 |- (((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))))
18	16, 17	ax-mp 7	. . . . . . 7 |- (((F |` A) |` (A \ B)):(A \ B)-onto->((F |` A)"(A \ B)) <-> (F |` (A \ B)):(A \ B)-onto->((F |` A)"(A \ B)))
19	16	rneqi 4187	. . . . . . . . 9 |- ran ((F |` A) |` (A \ B)) = ran ( F |` (A \ B))
20		df-ima 4007	. . . . . . . . 9 |- ((F |` A)"(A \ B)) = ran ((F |` A) |` (A \ B))
21		df-ima 4007	. . . . . . . . 9 |- (F"(A \ B)) = ran ( F |` (A \ B))
22	19, 20, 21	3eqtr4i 1921	. . . . . . . 8 |- ((F |` A)"(A \ B)) = (F"(A \ B))
23		foeq3 4615	. . . . . . . 8 |- (((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))))
24	22, 23	ax-mp 7	. . . . . . 7 |- ((F |` (A \ B)):(A \ B)-onto->((F |` A)"(A \ B)) <-> (F |` (A \ B)):(A \ B)-onto->(F"(A \ B)))
25	18, 24	bitri 190	. . . . . 6 |- (((F |` A) |` (A \ B)):(A \ B)-onto->((F |` A)"(A \ B)) <-> (F |` (A \ B)):(A \ B)-onto->(F"(A \ B)))
26	11, 25	sylib 215	. . . . 5 |- ((F |` A):A-onto->C -> (F |` (A \ B)):(A \ B)-onto->(F"(A \ B)))
27		funres11 4486	. . . . 5 |- (Fun `'F -> Fun `'(F |` (A \ B)))
28	2, 26, 27	syl2an 503	. . . 4 |- (((F |` A):A-onto->C /\ Fun `'F) -> (F |` (A \ B)):(A \ B)-1-1-onto->(F"(A \ B)))
29	28	ancoms 484	. . 3 |- ((Fun `'F /\ (F |` A):A-onto->C) -> (F |` (A \ B)):(A \ B)-1-1-onto->(F"(A \ B)))
30	29	3adant3 896	. 2 |- ((Fun `'F /\ (F |` A):A-onto->C /\ (F |` B):B-onto->D) -> (F |` (A \ B)):(A \ B)-1-1-onto->(F"(A \ B)))
31		imadif 4493	. . . . . 6 |- (Fun `'F -> (F"(A \ B)) = ((F"A) \ (F"B)))
32		difeq1 2717	. . . . . . 7 |- ((F"A) = C -> ((F"A) \ (F"B)) = (C \ (F"B)))
33		difeq2 2719	. . . . . . 7 |- ((F"B) = D -> (C \ (F"B)) = (C \ D))
34	32, 33	sylan9eq 1948	. . . . . 6 |- (((F"A) = C /\ (F"B) = D) -> ((F"A) \ (F"B)) = (C \ D))
35	31, 34	sylan9eq 1948	. . . . 5 |- ((Fun `'F /\ ((F"A) = C /\ (F"B) = D)) -> (F"(A \ B)) = (C \ D))
36		forn 4620	. . . . . . 7 |- ((F |` A):A-onto->C -> ran ( F |` A) = C)
37		df-ima 4007	. . . . . . 7 |- (F"A) = ran ( F |` A)
38	36, 37	syl5eq 1940	. . . . . 6 |- ((F |` A):A-onto->C -> (F"A) = C)
39		forn 4620	. . . . . . 7 |- ((F |` B):B-onto->D -> ran ( F |` B) = D)
40		df-ima 4007	. . . . . . 7 |- (F"B) = ran ( F |` B)
41	39, 40	syl5eq 1940	. . . . . 6 |- ((F |` B):B-onto->D -> (F"B) = D)
42	38, 41	anim12i 360	. . . . 5 |- (((F |` A):A-onto->C /\ (F |` B):B-onto->D) -> ((F"A) = C /\ (F"B) = D))
43	35, 42	sylan2 500	. . . 4 |- ((Fun `'F /\ ((F |` A):A-onto->C /\ (F |` B):B-onto->D)) -> (F"(A \ B)) = (C \ D))
44	43	3impb 1063	. . 3 |- ((Fun `'F /\ (F |` A):A-onto->C /\ (F |` B):B-onto->D) -> (F"(A \ B)) = (C \ D))
45		f1oeq3 4632	. . 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)))
46	44, 45	syl 12	. 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)))
47	30, 46	mpbid 212	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 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   \ cdif 2590   i^i cin 2592   C_ wss 2593  `'ccnv 3985  dom cdm 3986  ran crn 3987   |` cres 3988  "cima 3989  Fun wfun 3992  -->wf 3994  -onto->wfo 3996  -1-1-onto->wf1o 3997

This theorem is referenced by:  resin 4660

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013

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