Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5107 | . . . . . 6 | |
2 | difss 3070 | . . . . . . 7 | |
3 | fof 5106 | . . . . . . . 8 | |
4 | fdm 5050 | . . . . . . . 8 | |
5 | 3, 4 | syl 14 | . . . . . . 7 |
6 | 2, 5 | syl5sseqr 2994 | . . . . . 6 |
7 | fores 5115 | . . . . . 6 | |
8 | 1, 6, 7 | syl2anc 391 | . . . . 5 |
9 | resres 4624 | . . . . . . . 8 | |
10 | indif 3180 | . . . . . . . . 9 | |
11 | 10 | reseq2i 4609 | . . . . . . . 8 |
12 | 9, 11 | eqtri 2060 | . . . . . . 7 |
13 | foeq1 5102 | . . . . . . 7 | |
14 | 12, 13 | ax-mp 7 | . . . . . 6 |
15 | 12 | rneqi 4562 | . . . . . . . 8 |
16 | df-ima 4358 | . . . . . . . 8 | |
17 | df-ima 4358 | . . . . . . . 8 | |
18 | 15, 16, 17 | 3eqtr4i 2070 | . . . . . . 7 |
19 | foeq3 5104 | . . . . . . 7 | |
20 | 18, 19 | ax-mp 7 | . . . . . 6 |
21 | 14, 20 | bitri 173 | . . . . 5 |
22 | 8, 21 | sylib 127 | . . . 4 |
23 | funres11 4971 | . . . 4 | |
24 | dff1o3 5132 | . . . . 5 | |
25 | 24 | biimpri 124 | . . . 4 |
26 | 22, 23, 25 | syl2anr 274 | . . 3 |
27 | 26 | 3adant3 924 | . 2 |
28 | df-ima 4358 | . . . . . . 7 | |
29 | forn 5109 | . . . . . . 7 | |
30 | 28, 29 | syl5eq 2084 | . . . . . 6 |
31 | df-ima 4358 | . . . . . . 7 | |
32 | forn 5109 | . . . . . . 7 | |
33 | 31, 32 | syl5eq 2084 | . . . . . 6 |
34 | 30, 33 | anim12i 321 | . . . . 5 |
35 | imadif 4979 | . . . . . 6 | |
36 | difeq12 3057 | . . . . . 6 | |
37 | 35, 36 | sylan9eq 2092 | . . . . 5 |
38 | 34, 37 | sylan2 270 | . . . 4 |
39 | 38 | 3impb 1100 | . . 3 |
40 | f1oeq3 5119 | . . 3 | |
41 | 39, 40 | syl 14 | . 2 |
42 | 27, 41 | mpbid 135 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 cdif 2914 cin 2916 wss 2917 ccnv 4344 cdm 4345 crn 4346 cres 4347 cima 4348 wfun 4896 wf 4898 wfo 4900 wf1o 4901 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 |
This theorem is referenced by: dif1en 6337 |
Copyright terms: Public domain | W3C validator |