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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 5264 | . . . . . . . 8 | |
2 | velsn 3514 | . . . . . . . . 9 | |
3 | velsn 3514 | . . . . . . . . 9 | |
4 | 2, 3 | anbi12i 455 | . . . . . . . 8 |
5 | 1, 4 | sylib 121 | . . . . . . 7 |
6 | 5 | ex 114 | . . . . . 6 |
7 | fsn.1 | . . . . . . . . . 10 | |
8 | 7 | snid 3526 | . . . . . . . . 9 |
9 | feu 5275 | . . . . . . . . 9 | |
10 | 8, 9 | mpan2 421 | . . . . . . . 8 |
11 | 3 | anbi1i 453 | . . . . . . . . . . 11 |
12 | opeq2 3676 | . . . . . . . . . . . . . 14 | |
13 | 12 | eleq1d 2186 | . . . . . . . . . . . . 13 |
14 | 13 | pm5.32i 449 | . . . . . . . . . . . 12 |
15 | ancom 264 | . . . . . . . . . . . 12 | |
16 | 14, 15 | bitr4i 186 | . . . . . . . . . . 11 |
17 | 11, 16 | bitr2i 184 | . . . . . . . . . 10 |
18 | 17 | eubii 1986 | . . . . . . . . 9 |
19 | fsn.2 | . . . . . . . . . . . 12 | |
20 | 19 | eueq1 2829 | . . . . . . . . . . 11 |
21 | 20 | biantru 300 | . . . . . . . . . 10 |
22 | euanv 2034 | . . . . . . . . . 10 | |
23 | 21, 22 | bitr4i 186 | . . . . . . . . 9 |
24 | df-reu 2400 | . . . . . . . . 9 | |
25 | 18, 23, 24 | 3bitr4i 211 | . . . . . . . 8 |
26 | 10, 25 | sylibr 133 | . . . . . . 7 |
27 | opeq12 3677 | . . . . . . . 8 | |
28 | 27 | eleq1d 2186 | . . . . . . 7 |
29 | 26, 28 | syl5ibrcom 156 | . . . . . 6 |
30 | 6, 29 | impbid 128 | . . . . 5 |
31 | vex 2663 | . . . . . . . 8 | |
32 | vex 2663 | . . . . . . . 8 | |
33 | 31, 32 | opex 4121 | . . . . . . 7 |
34 | 33 | elsn 3513 | . . . . . 6 |
35 | 7, 19 | opth2 4132 | . . . . . 6 |
36 | 34, 35 | bitr2i 184 | . . . . 5 |
37 | 30, 36 | syl6bb 195 | . . . 4 |
38 | 37 | alrimivv 1831 | . . 3 |
39 | frel 5247 | . . . 4 | |
40 | 7, 19 | relsnop 4615 | . . . 4 |
41 | eqrel 4598 | . . . 4 | |
42 | 39, 40, 41 | sylancl 409 | . . 3 |
43 | 38, 42 | mpbird 166 | . 2 |
44 | 7, 19 | f1osn 5375 | . . . 4 |
45 | f1oeq1 5326 | . . . 4 | |
46 | 44, 45 | mpbiri 167 | . . 3 |
47 | f1of 5335 | . . 3 | |
48 | 46, 47 | syl 14 | . 2 |
49 | 43, 48 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wal 1314 wceq 1316 wcel 1465 weu 1977 wreu 2395 cvv 2660 csn 3497 cop 3500 wrel 4514 wf 5089 wf1o 5092 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 |
This theorem is referenced by: fsng 5561 mapsn 6552 |
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