Theorem resin 4660

Description: The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.)

Assertion

Ref	Expression
resin	|- ((Fun `'F /\ (F |` A):A-onto->C /\ (F |` B):B-onto->D) -> (F |` (A i^i B)):(A i^i B)-1-1-onto->(C i^i D))

Proof of Theorem resin

Step	Hyp	Ref	Expression
1		resdif 4659	. . 3 |- ((Fun `'F /\ (F |` A):A-onto->C /\ (F |` (A \ B)):(A \ B)-onto->(C \ D)) -> (F |` (A \ (A \ B))):(A \ (A \ B))-1-1-onto->(C \ (C \ D)))
2		resdif 4659	. . . 4 |- ((Fun `'F /\ (F |` A):A-onto->C /\ (F |` B):B-onto->D) -> (F |` (A \ B)):(A \ B)-1-1-onto->(C \ D))
3		f1ofo 4643	. . . 4 |- ((F |` (A \ B)):(A \ B)-1-1-onto->(C \ D) -> (F |` (A \ B)):(A \ B)-onto->(C \ D))
4	2, 3	syl 12	. . 3 |- ((Fun `'F /\ (F |` A):A-onto->C /\ (F |` B):B-onto->D) -> (F |` (A \ B)):(A \ B)-onto->(C \ D))
5	1, 4	syld3an3 1142	. 2 |- ((Fun `'F /\ (F |` A):A-onto->C /\ (F |` B):B-onto->D) -> (F |` (A \ (A \ B))):(A \ (A \ B))-1-1-onto->(C \ (C \ D)))
6		dfin4 2835	. . . 4 |- (C i^i D) = (C \ (C \ D))
7		f1oeq3 4632	. . . 4 |- ((C i^i D) = (C \ (C \ D)) -> ((F |` (A i^i B)):(A i^i B)-1-1-onto->(C i^i D) <-> (F |` (A i^i B)):(A i^i B)-1-1-onto->(C \ (C \ D))))
8	6, 7	ax-mp 7	. . 3 |- ((F |` (A i^i B)):(A i^i B)-1-1-onto->(C i^i D) <-> (F |` (A i^i B)):(A i^i B)-1-1-onto->(C \ (C \ D)))
9		dfin4 2835	. . . 4 |- (A i^i B) = (A \ (A \ B))
10		f1oeq2 4631	. . . 4 |- ((A i^i B) = (A \ (A \ B)) -> ((F |` (A i^i B)):(A i^i B)-1-1-onto->(C \ (C \ D)) <-> (F |` (A i^i B)):(A \ (A \ B))-1-1-onto->(C \ (C \ D))))
11	9, 10	ax-mp 7	. . 3 |- ((F |` (A i^i B)):(A i^i B)-1-1-onto->(C \ (C \ D)) <-> (F |` (A i^i B)):(A \ (A \ B))-1-1-onto->(C \ (C \ D)))
12		reseq2 4219	. . . . 5 |- ((A i^i B) = (A \ (A \ B)) -> (F |` (A i^i B)) = (F |` (A \ (A \ B))))
13	9, 12	ax-mp 7	. . . 4 |- (F |` (A i^i B)) = (F |` (A \ (A \ B)))
14		f1oeq1 4630	. . . 4 |- ((F |` (A i^i B)) = (F |` (A \ (A \ B))) -> ((F |` (A i^i B)):(A \ (A \ B))-1-1-onto->(C \ (C \ D)) <-> (F |` (A \ (A \ B))):(A \ (A \ B))-1-1-onto->(C \ (C \ D))))
15	13, 14	ax-mp 7	. . 3 |- ((F |` (A i^i B)):(A \ (A \ B))-1-1-onto->(C \ (C \ D)) <-> (F |` (A \ (A \ B))):(A \ (A \ B))-1-1-onto->(C \ (C \ D)))
16	8, 11, 15	3bitrri 195	. 2 |- ((F |` (A \ (A \ B))):(A \ (A \ B))-1-1-onto->(C \ (C \ D)) <-> (F |` (A i^i B)):(A i^i B)-1-1-onto->(C i^i D))
17	5, 16	sylib 215	1 |- ((Fun `'F /\ (F |` A):A-onto->C /\ (F |` B):B-onto->D) -> (F |` (A i^i B)):(A i^i B)-1-1-onto->(C i^i D))

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858   = wceq 1298   \ cdif 2590   i^i cin 2592  `'ccnv 3985   |` cres 3988  Fun wfun 3992  -onto->wfo 3996  -1-1-onto->wf1o 3997

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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