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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 |
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fsn.2 |
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Ref | Expression |
---|---|
fsn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelf 5383 |
. . . . . . . 8
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2 | velsn 3608 |
. . . . . . . . 9
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3 | velsn 3608 |
. . . . . . . . 9
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4 | 2, 3 | anbi12i 460 |
. . . . . . . 8
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5 | 1, 4 | sylib 122 |
. . . . . . 7
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6 | 5 | ex 115 |
. . . . . 6
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7 | fsn.1 |
. . . . . . . . . 10
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8 | 7 | snid 3622 |
. . . . . . . . 9
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9 | feu 5394 |
. . . . . . . . 9
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10 | 8, 9 | mpan2 425 |
. . . . . . . 8
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11 | 3 | anbi1i 458 |
. . . . . . . . . . 11
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12 | opeq2 3777 |
. . . . . . . . . . . . . 14
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13 | 12 | eleq1d 2246 |
. . . . . . . . . . . . 13
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14 | 13 | pm5.32i 454 |
. . . . . . . . . . . 12
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15 | ancom 266 |
. . . . . . . . . . . 12
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16 | 14, 15 | bitr4i 187 |
. . . . . . . . . . 11
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17 | 11, 16 | bitr2i 185 |
. . . . . . . . . 10
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18 | 17 | eubii 2035 |
. . . . . . . . 9
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19 | fsn.2 |
. . . . . . . . . . . 12
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20 | 19 | eueq1 2909 |
. . . . . . . . . . 11
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21 | 20 | biantru 302 |
. . . . . . . . . 10
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22 | euanv 2083 |
. . . . . . . . . 10
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23 | 21, 22 | bitr4i 187 |
. . . . . . . . 9
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24 | df-reu 2462 |
. . . . . . . . 9
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25 | 18, 23, 24 | 3bitr4i 212 |
. . . . . . . 8
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26 | 10, 25 | sylibr 134 |
. . . . . . 7
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27 | opeq12 3778 |
. . . . . . . 8
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28 | 27 | eleq1d 2246 |
. . . . . . 7
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29 | 26, 28 | syl5ibrcom 157 |
. . . . . 6
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30 | 6, 29 | impbid 129 |
. . . . 5
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31 | vex 2740 |
. . . . . . . 8
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32 | vex 2740 |
. . . . . . . 8
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33 | 31, 32 | opex 4226 |
. . . . . . 7
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34 | 33 | elsn 3607 |
. . . . . 6
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35 | 7, 19 | opth2 4237 |
. . . . . 6
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36 | 34, 35 | bitr2i 185 |
. . . . 5
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37 | 30, 36 | bitrdi 196 |
. . . 4
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38 | 37 | alrimivv 1875 |
. . 3
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39 | frel 5366 |
. . . 4
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40 | 7, 19 | relsnop 4729 |
. . . 4
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41 | eqrel 4712 |
. . . 4
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42 | 39, 40, 41 | sylancl 413 |
. . 3
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43 | 38, 42 | mpbird 167 |
. 2
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44 | 7, 19 | f1osn 5497 |
. . . 4
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45 | f1oeq1 5445 |
. . . 4
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46 | 44, 45 | mpbiri 168 |
. . 3
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47 | f1of 5457 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
48 | 46, 47 | syl 14 |
. 2
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49 | 43, 48 | impbii 126 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 |
This theorem is referenced by: fsng 5685 mapsn 6684 |
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