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Mirrors > Home > ILE Home > Th. List > elixpsn | Unicode version |
Description: Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
elixpsn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3477 |
. . . 4
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2 | 1 | ixpeq1d 6507 |
. . 3
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3 | 2 | eleq2d 2164 |
. 2
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4 | opeq1 3644 |
. . . . 5
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5 | 4 | sneqd 3479 |
. . . 4
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6 | 5 | eqeq2d 2106 |
. . 3
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7 | 6 | rexbidv 2392 |
. 2
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8 | elex 2644 |
. . 3
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9 | vex 2636 |
. . . . . . 7
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10 | vex 2636 |
. . . . . . 7
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11 | 9, 10 | opex 4080 |
. . . . . 6
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12 | 11 | snex 4041 |
. . . . 5
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13 | eleq1 2157 |
. . . . 5
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14 | 12, 13 | mpbiri 167 |
. . . 4
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15 | 14 | rexlimivw 2498 |
. . 3
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16 | eleq1 2157 |
. . . 4
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17 | eqeq1 2101 |
. . . . 5
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18 | 17 | rexbidv 2392 |
. . . 4
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19 | vex 2636 |
. . . . . 6
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20 | 19 | elixp 6502 |
. . . . 5
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21 | fveq2 5340 |
. . . . . . . 8
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22 | 21 | eleq1d 2163 |
. . . . . . 7
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23 | 9, 22 | ralsn 3506 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23 | anbi2i 446 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | simpl 108 |
. . . . . . . . 9
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26 | fveq2 5340 |
. . . . . . . . . . . . 13
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27 | 26 | eleq1d 2163 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 9, 27 | ralsn 3506 |
. . . . . . . . . . 11
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29 | 28 | biimpri 132 |
. . . . . . . . . 10
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30 | 29 | adantl 272 |
. . . . . . . . 9
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31 | ffnfv 5495 |
. . . . . . . . 9
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32 | 25, 30, 31 | sylanbrc 409 |
. . . . . . . 8
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33 | 9 | fsn2 5510 |
. . . . . . . 8
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34 | 32, 33 | sylib 121 |
. . . . . . 7
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35 | opeq2 3645 |
. . . . . . . . 9
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36 | 35 | sneqd 3479 |
. . . . . . . 8
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37 | 36 | rspceeqv 2753 |
. . . . . . 7
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38 | 34, 37 | syl 14 |
. . . . . 6
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39 | 9, 10 | fvsn 5531 |
. . . . . . . . . 10
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40 | id 19 |
. . . . . . . . . 10
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41 | 39, 40 | syl5eqel 2181 |
. . . . . . . . 9
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42 | 9, 10 | fnsn 5102 |
. . . . . . . . 9
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43 | 41, 42 | jctil 306 |
. . . . . . . 8
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44 | fneq1 5136 |
. . . . . . . . 9
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45 | fveq1 5339 |
. . . . . . . . . 10
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46 | 45 | eleq1d 2163 |
. . . . . . . . 9
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47 | 44, 46 | anbi12d 458 |
. . . . . . . 8
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48 | 43, 47 | syl5ibrcom 156 |
. . . . . . 7
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49 | 48 | rexlimiv 2496 |
. . . . . 6
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50 | 38, 49 | impbii 125 |
. . . . 5
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51 | 20, 24, 50 | 3bitri 205 |
. . . 4
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52 | 16, 18, 51 | vtoclbg 2694 |
. . 3
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53 | 8, 15, 52 | pm5.21nii 658 |
. 2
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54 | 3, 7, 53 | vtoclbg 2694 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-reu 2377 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-ixp 6496 |
This theorem is referenced by: ixpsnf1o 6533 |
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