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Mirrors > Home > ILE Home > Th. List > fsn2 | Unicode version |
Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.) |
Ref | Expression |
---|---|
fsn2.1 |
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Ref | Expression |
---|---|
fsn2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5361 |
. . 3
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2 | fsn2.1 |
. . . . 5
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3 | 2 | snid 3622 |
. . . 4
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4 | funfvex 5528 |
. . . . 5
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5 | 4 | funfni 5312 |
. . . 4
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6 | 3, 5 | mpan2 425 |
. . 3
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7 | 1, 6 | syl 14 |
. 2
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8 | elex 2748 |
. . 3
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9 | 8 | adantr 276 |
. 2
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10 | ffvelcdm 5645 |
. . . . . 6
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11 | 3, 10 | mpan2 425 |
. . . . 5
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12 | dffn3 5372 |
. . . . . . . 8
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13 | 12 | biimpi 120 |
. . . . . . 7
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14 | imadmrn 4976 |
. . . . . . . . . 10
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15 | fndm 5311 |
. . . . . . . . . . 11
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16 | 15 | imaeq2d 4966 |
. . . . . . . . . 10
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17 | 14, 16 | eqtr3id 2224 |
. . . . . . . . 9
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18 | fnsnfv 5571 |
. . . . . . . . . 10
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19 | 3, 18 | mpan2 425 |
. . . . . . . . 9
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20 | 17, 19 | eqtr4d 2213 |
. . . . . . . 8
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21 | feq3 5346 |
. . . . . . . 8
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22 | 20, 21 | syl 14 |
. . . . . . 7
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23 | 13, 22 | mpbid 147 |
. . . . . 6
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24 | 1, 23 | syl 14 |
. . . . 5
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25 | 11, 24 | jca 306 |
. . . 4
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26 | snssi 3735 |
. . . . 5
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27 | fss 5373 |
. . . . . 6
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28 | 27 | ancoms 268 |
. . . . 5
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29 | 26, 28 | sylan 283 |
. . . 4
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30 | 25, 29 | impbii 126 |
. . 3
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31 | fsng 5685 |
. . . . 5
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32 | 2, 31 | mpan 424 |
. . . 4
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33 | 32 | anbi2d 464 |
. . 3
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34 | 30, 33 | bitrid 192 |
. 2
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35 | 7, 9, 34 | pm5.21nii 704 |
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-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 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-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 |
This theorem is referenced by: fnressn 5698 fressnfv 5699 mapsnconst 6688 elixpsn 6729 en1 6793 |
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