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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 5341 | . . . . . 6 | |
2 | difss 3197 | . . . . . . 7 | |
3 | fof 5340 | . . . . . . . 8 | |
4 | fdm 5273 | . . . . . . . 8 | |
5 | 3, 4 | syl 14 | . . . . . . 7 |
6 | 2, 5 | sseqtrrid 3143 | . . . . . 6 |
7 | fores 5349 | . . . . . 6 | |
8 | 1, 6, 7 | syl2anc 408 | . . . . 5 |
9 | resres 4826 | . . . . . . . 8 | |
10 | indif 3314 | . . . . . . . . 9 | |
11 | 10 | reseq2i 4811 | . . . . . . . 8 |
12 | 9, 11 | eqtri 2158 | . . . . . . 7 |
13 | foeq1 5336 | . . . . . . 7 | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 |
15 | 12 | rneqi 4762 | . . . . . . . 8 |
16 | df-ima 4547 | . . . . . . . 8 | |
17 | df-ima 4547 | . . . . . . . 8 | |
18 | 15, 16, 17 | 3eqtr4i 2168 | . . . . . . 7 |
19 | foeq3 5338 | . . . . . . 7 | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 |
21 | 14, 20 | bitri 183 | . . . . 5 |
22 | 8, 21 | sylib 121 | . . . 4 |
23 | funres11 5190 | . . . 4 | |
24 | dff1o3 5366 | . . . . 5 | |
25 | 24 | biimpri 132 | . . . 4 |
26 | 22, 23, 25 | syl2anr 288 | . . 3 |
27 | 26 | 3adant3 1001 | . 2 |
28 | df-ima 4547 | . . . . . . 7 | |
29 | forn 5343 | . . . . . . 7 | |
30 | 28, 29 | syl5eq 2182 | . . . . . 6 |
31 | df-ima 4547 | . . . . . . 7 | |
32 | forn 5343 | . . . . . . 7 | |
33 | 31, 32 | syl5eq 2182 | . . . . . 6 |
34 | 30, 33 | anim12i 336 | . . . . 5 |
35 | imadif 5198 | . . . . . 6 | |
36 | difeq12 3184 | . . . . . 6 | |
37 | 35, 36 | sylan9eq 2190 | . . . . 5 |
38 | 34, 37 | sylan2 284 | . . . 4 |
39 | 38 | 3impb 1177 | . . 3 |
40 | f1oeq3 5353 | . . 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 962 wceq 1331 cdif 3063 cin 3065 wss 3066 ccnv 4533 cdm 4534 crn 4535 cres 4536 cima 4537 wfun 5112 wf 5114 wfo 5116 wf1o 5117 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 |
This theorem is referenced by: dif1en 6766 |
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