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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 5127 | . . . . . 6 | |
2 | difss 3098 | . . . . . . 7 | |
3 | fof 5126 | . . . . . . . 8 | |
4 | fdm 5070 | . . . . . . . 8 | |
5 | 3, 4 | syl 14 | . . . . . . 7 |
6 | 2, 5 | syl5sseqr 3048 | . . . . . 6 |
7 | fores 5135 | . . . . . 6 | |
8 | 1, 6, 7 | syl2anc 403 | . . . . 5 |
9 | resres 4642 | . . . . . . . 8 | |
10 | indif 3207 | . . . . . . . . 9 | |
11 | 10 | reseq2i 4627 | . . . . . . . 8 |
12 | 9, 11 | eqtri 2101 | . . . . . . 7 |
13 | foeq1 5122 | . . . . . . 7 | |
14 | 12, 13 | ax-mp 7 | . . . . . 6 |
15 | 12 | rneqi 4580 | . . . . . . . 8 |
16 | df-ima 4376 | . . . . . . . 8 | |
17 | df-ima 4376 | . . . . . . . 8 | |
18 | 15, 16, 17 | 3eqtr4i 2111 | . . . . . . 7 |
19 | foeq3 5124 | . . . . . . 7 | |
20 | 18, 19 | ax-mp 7 | . . . . . 6 |
21 | 14, 20 | bitri 182 | . . . . 5 |
22 | 8, 21 | sylib 120 | . . . 4 |
23 | funres11 4991 | . . . 4 | |
24 | dff1o3 5152 | . . . . 5 | |
25 | 24 | biimpri 131 | . . . 4 |
26 | 22, 23, 25 | syl2anr 284 | . . 3 |
27 | 26 | 3adant3 958 | . 2 |
28 | df-ima 4376 | . . . . . . 7 | |
29 | forn 5129 | . . . . . . 7 | |
30 | 28, 29 | syl5eq 2125 | . . . . . 6 |
31 | df-ima 4376 | . . . . . . 7 | |
32 | forn 5129 | . . . . . . 7 | |
33 | 31, 32 | syl5eq 2125 | . . . . . 6 |
34 | 30, 33 | anim12i 331 | . . . . 5 |
35 | imadif 4999 | . . . . . 6 | |
36 | difeq12 3085 | . . . . . 6 | |
37 | 35, 36 | sylan9eq 2133 | . . . . 5 |
38 | 34, 37 | sylan2 280 | . . . 4 |
39 | 38 | 3impb 1134 | . . 3 |
40 | f1oeq3 5139 | . . 3 | |
41 | 39, 40 | syl 14 | . 2 |
42 | 27, 41 | mpbid 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 cdif 2970 cin 2972 wss 2973 ccnv 4362 cdm 4363 crn 4364 cres 4365 cima 4366 wfun 4916 wf 4918 wfo 4920 wf1o 4921 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 |
This theorem is referenced by: dif1en 6364 |
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