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Mirrors > Home > ILE Home > Th. List > resdif | Unicode version |
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.) |
Ref | Expression |
---|---|
resdif |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofun 5405 | . . . . . 6 | |
2 | difss 3243 | . . . . . . 7 | |
3 | fof 5404 | . . . . . . . 8 | |
4 | fdm 5337 | . . . . . . . 8 | |
5 | 3, 4 | syl 14 | . . . . . . 7 |
6 | 2, 5 | sseqtrrid 3188 | . . . . . 6 |
7 | fores 5413 | . . . . . 6 | |
8 | 1, 6, 7 | syl2anc 409 | . . . . 5 |
9 | resres 4890 | . . . . . . . 8 | |
10 | indif 3360 | . . . . . . . . 9 | |
11 | 10 | reseq2i 4875 | . . . . . . . 8 |
12 | 9, 11 | eqtri 2185 | . . . . . . 7 |
13 | foeq1 5400 | . . . . . . 7 | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 |
15 | 12 | rneqi 4826 | . . . . . . . 8 |
16 | df-ima 4611 | . . . . . . . 8 | |
17 | df-ima 4611 | . . . . . . . 8 | |
18 | 15, 16, 17 | 3eqtr4i 2195 | . . . . . . 7 |
19 | foeq3 5402 | . . . . . . 7 | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 |
21 | 14, 20 | bitri 183 | . . . . 5 |
22 | 8, 21 | sylib 121 | . . . 4 |
23 | funres11 5254 | . . . 4 | |
24 | dff1o3 5432 | . . . . 5 | |
25 | 24 | biimpri 132 | . . . 4 |
26 | 22, 23, 25 | syl2anr 288 | . . 3 |
27 | 26 | 3adant3 1006 | . 2 |
28 | df-ima 4611 | . . . . . . 7 | |
29 | forn 5407 | . . . . . . 7 | |
30 | 28, 29 | syl5eq 2209 | . . . . . 6 |
31 | df-ima 4611 | . . . . . . 7 | |
32 | forn 5407 | . . . . . . 7 | |
33 | 31, 32 | syl5eq 2209 | . . . . . 6 |
34 | 30, 33 | anim12i 336 | . . . . 5 |
35 | imadif 5262 | . . . . . 6 | |
36 | difeq12 3230 | . . . . . 6 | |
37 | 35, 36 | sylan9eq 2217 | . . . . 5 |
38 | 34, 37 | sylan2 284 | . . . 4 |
39 | 38 | 3impb 1188 | . . 3 |
40 | f1oeq3 5417 | . . 3 | |
41 | 39, 40 | syl 14 | . 2 |
42 | 27, 41 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 cdif 3108 cin 3110 wss 3111 ccnv 4597 cdm 4598 crn 4599 cres 4600 cima 4601 wfun 5176 wf 5178 wfo 5180 wf1o 5181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 |
This theorem is referenced by: dif1en 6836 |
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