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Theorem resdif 5306

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	$/\$ $/\$

**Proof of Theorem resdif**
Step	Hyp	Ref	Expression
1		fofun 5270	. . . . . 6
2		difss 3393	. . . . . . 7
3		fof 5269	. . . . . . . 8
4		fdm 5226	. . . . . . . 8
5	3, 4	syl 15	. . . . . . 7
6	2, 5	syl5sseqr 3320	. . . . . 6
7		fores 5278	. . . . . 6 $/\$
8	1, 6, 7	syl2anc 642	. . . . 5
9		resabs1 4992	. . . . . . . 8
10	2, 9	ax-mp 8	. . . . . . 7
11		foeq1 5265	. . . . . . 7
12	10, 11	ax-mp 8	. . . . . 6
13	10	rneqi 4957	. . . . . . . 8
14		dfima3 4951	. . . . . . . 8
15		dfima3 4951	. . . . . . . 8
16	13, 14, 15	3eqtr4i 2383	. . . . . . 7
17		foeq3 5267	. . . . . . 7
18	16, 17	ax-mp 8	. . . . . 6
19	12, 18	bitri 240	. . . . 5
20	8, 19	sylib 188	. . . 4
21		funres11 5164	. . . 4
22		dff1o3 5292	. . . . 5 $/\$
23	22	biimpri 197	. . . 4 $/\$
24	20, 21, 23	syl2anr 464	. . 3 $/\$
25	24	3adant3 975	. 2 $/\$ $/\$
26		dfima3 4951	. . . . . . 7
27		forn 5272	. . . . . . 7
28	26, 27	syl5eq 2397	. . . . . 6
29		dfima3 4951	. . . . . . 7
30		forn 5272	. . . . . . 7
31	29, 30	syl5eq 2397	. . . . . 6
32	28, 31	anim12i 549	. . . . 5 $/\$ $/\$
33		imadif 5171	. . . . . 6
34		difeq12 3380	. . . . . 6 $/\$
35	33, 34	sylan9eq 2405	. . . . 5 $/\$ $/\$
36	32, 35	sylan2 460	. . . 4 $/\$ $/\$
37	36	3impb 1147	. . 3 $/\$ $/\$
38		f1oeq3 5283	. . 3
39	37, 38	syl 15	. 2 $/\$ $/\$
40	25, 39	mpbid 201	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 176 $/\$ wa 358 $/\$ w3a 934

wceq 1642

cdif 3206

wss 3257

cima 4722

ccnv 4771

cdm 4772

crn 4773

cres 4774

wfun 4775

wf 4777

wfo 4779

wf1o 4780

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794

This theorem is referenced by: resin 5307

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