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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 5379 | . . . . . . . 8 | |
2 | velsn 3606 | . . . . . . . . 9 | |
3 | velsn 3606 | . . . . . . . . 9 | |
4 | 2, 3 | anbi12i 460 | . . . . . . . 8 |
5 | 1, 4 | sylib 122 | . . . . . . 7 |
6 | 5 | ex 115 | . . . . . 6 |
7 | fsn.1 | . . . . . . . . . 10 | |
8 | 7 | snid 3620 | . . . . . . . . 9 |
9 | feu 5390 | . . . . . . . . 9 | |
10 | 8, 9 | mpan2 425 | . . . . . . . 8 |
11 | 3 | anbi1i 458 | . . . . . . . . . . 11 |
12 | opeq2 3775 | . . . . . . . . . . . . . 14 | |
13 | 12 | eleq1d 2244 | . . . . . . . . . . . . 13 |
14 | 13 | pm5.32i 454 | . . . . . . . . . . . 12 |
15 | ancom 266 | . . . . . . . . . . . 12 | |
16 | 14, 15 | bitr4i 187 | . . . . . . . . . . 11 |
17 | 11, 16 | bitr2i 185 | . . . . . . . . . 10 |
18 | 17 | eubii 2033 | . . . . . . . . 9 |
19 | fsn.2 | . . . . . . . . . . . 12 | |
20 | 19 | eueq1 2907 | . . . . . . . . . . 11 |
21 | 20 | biantru 302 | . . . . . . . . . 10 |
22 | euanv 2081 | . . . . . . . . . 10 | |
23 | 21, 22 | bitr4i 187 | . . . . . . . . 9 |
24 | df-reu 2460 | . . . . . . . . 9 | |
25 | 18, 23, 24 | 3bitr4i 212 | . . . . . . . 8 |
26 | 10, 25 | sylibr 134 | . . . . . . 7 |
27 | opeq12 3776 | . . . . . . . 8 | |
28 | 27 | eleq1d 2244 | . . . . . . 7 |
29 | 26, 28 | syl5ibrcom 157 | . . . . . 6 |
30 | 6, 29 | impbid 129 | . . . . 5 |
31 | vex 2738 | . . . . . . . 8 | |
32 | vex 2738 | . . . . . . . 8 | |
33 | 31, 32 | opex 4223 | . . . . . . 7 |
34 | 33 | elsn 3605 | . . . . . 6 |
35 | 7, 19 | opth2 4234 | . . . . . 6 |
36 | 34, 35 | bitr2i 185 | . . . . 5 |
37 | 30, 36 | bitrdi 196 | . . . 4 |
38 | 37 | alrimivv 1873 | . . 3 |
39 | frel 5362 | . . . 4 | |
40 | 7, 19 | relsnop 4726 | . . . 4 |
41 | eqrel 4709 | . . . 4 | |
42 | 39, 40, 41 | sylancl 413 | . . 3 |
43 | 38, 42 | mpbird 167 | . 2 |
44 | 7, 19 | f1osn 5493 | . . . 4 |
45 | f1oeq1 5441 | . . . 4 | |
46 | 44, 45 | mpbiri 168 | . . 3 |
47 | f1of 5453 | . . 3 | |
48 | 46, 47 | syl 14 | . 2 |
49 | 43, 48 | impbii 126 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 104 wb 105 wal 1351 wceq 1353 weu 2024 wcel 2146 wreu 2455 cvv 2735 csn 3589 cop 3592 wrel 4625 wf 5204 wf1o 5207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-reu 2460 df-v 2737 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-br 3999 df-opab 4060 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 |
This theorem is referenced by: fsng 5681 mapsn 6680 |
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