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Mirrors > Home > ILE Home > Th. List > fsn | Unicode version |
Description: A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.) |
Ref | Expression |
---|---|
fsn.1 | |
fsn.2 |
Ref | Expression |
---|---|
fsn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelf 5062 | . . . . . . . 8 | |
2 | velsn 3392 | . . . . . . . . 9 | |
3 | velsn 3392 | . . . . . . . . 9 | |
4 | 2, 3 | anbi12i 433 | . . . . . . . 8 |
5 | 1, 4 | sylib 127 | . . . . . . 7 |
6 | 5 | ex 108 | . . . . . 6 |
7 | fsn.1 | . . . . . . . . . 10 | |
8 | 7 | snid 3402 | . . . . . . . . 9 |
9 | feu 5072 | . . . . . . . . 9 | |
10 | 8, 9 | mpan2 401 | . . . . . . . 8 |
11 | 3 | anbi1i 431 | . . . . . . . . . . 11 |
12 | opeq2 3550 | . . . . . . . . . . . . . 14 | |
13 | 12 | eleq1d 2106 | . . . . . . . . . . . . 13 |
14 | 13 | pm5.32i 427 | . . . . . . . . . . . 12 |
15 | ancom 253 | . . . . . . . . . . . 12 | |
16 | 14, 15 | bitr4i 176 | . . . . . . . . . . 11 |
17 | 11, 16 | bitr2i 174 | . . . . . . . . . 10 |
18 | 17 | eubii 1909 | . . . . . . . . 9 |
19 | fsn.2 | . . . . . . . . . . . 12 | |
20 | 19 | eueq1 2713 | . . . . . . . . . . 11 |
21 | 20 | biantru 286 | . . . . . . . . . 10 |
22 | euanv 1957 | . . . . . . . . . 10 | |
23 | 21, 22 | bitr4i 176 | . . . . . . . . 9 |
24 | df-reu 2313 | . . . . . . . . 9 | |
25 | 18, 23, 24 | 3bitr4i 201 | . . . . . . . 8 |
26 | 10, 25 | sylibr 137 | . . . . . . 7 |
27 | opeq12 3551 | . . . . . . . 8 | |
28 | 27 | eleq1d 2106 | . . . . . . 7 |
29 | 26, 28 | syl5ibrcom 146 | . . . . . 6 |
30 | 6, 29 | impbid 120 | . . . . 5 |
31 | vex 2560 | . . . . . . . 8 | |
32 | vex 2560 | . . . . . . . 8 | |
33 | 31, 32 | opex 3966 | . . . . . . 7 |
34 | 33 | elsn 3391 | . . . . . 6 |
35 | 7, 19 | opth2 3977 | . . . . . 6 |
36 | 34, 35 | bitr2i 174 | . . . . 5 |
37 | 30, 36 | syl6bb 185 | . . . 4 |
38 | 37 | alrimivv 1755 | . . 3 |
39 | frel 5049 | . . . 4 | |
40 | 7, 19 | relsnop 4444 | . . . 4 |
41 | eqrel 4429 | . . . 4 | |
42 | 39, 40, 41 | sylancl 392 | . . 3 |
43 | 38, 42 | mpbird 156 | . 2 |
44 | 7, 19 | f1osn 5166 | . . . 4 |
45 | f1oeq1 5117 | . . . 4 | |
46 | 44, 45 | mpbiri 157 | . . 3 |
47 | f1of 5126 | . . 3 | |
48 | 46, 47 | syl 14 | . 2 |
49 | 43, 48 | impbii 117 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wal 1241 wceq 1243 wcel 1393 weu 1900 wreu 2308 cvv 2557 csn 3375 cop 3378 wrel 4350 wf 4898 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-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-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-reu 2313 df-v 2559 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-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 |
This theorem is referenced by: fsng 5336 |
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