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Mirrors > Home > ILE Home > Th. List > fnressn | Unicode version |
Description: A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.) |
Ref | Expression |
---|---|
fnressn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3603 |
. . . . . 6
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2 | 1 | reseq2d 4907 |
. . . . 5
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3 | fveq2 5515 |
. . . . . . 7
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4 | opeq12 3780 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 3, 4 | mpdan 421 |
. . . . . 6
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6 | 5 | sneqd 3605 |
. . . . 5
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7 | 2, 6 | eqeq12d 2192 |
. . . 4
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8 | 7 | imbi2d 230 |
. . 3
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9 | vex 2740 |
. . . . . . 7
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10 | 9 | snss 3727 |
. . . . . 6
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11 | fnssres 5329 |
. . . . . 6
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12 | 10, 11 | sylan2b 287 |
. . . . 5
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13 | dffn2 5367 |
. . . . . . 7
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14 | 9 | fsn2 5690 |
. . . . . . 7
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15 | 13, 14 | bitri 184 |
. . . . . 6
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16 | snssi 3736 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 16, 11 | sylan2 286 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | vsnid 3624 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() | |
19 | funfvex 5532 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19 | funfni 5316 |
. . . . . . . . 9
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21 | 17, 18, 20 | sylancl 413 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 21 | biantrurd 305 |
. . . . . . 7
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23 | fvres 5539 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 18, 23 | ax-mp 5 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | opeq2i 3782 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 25 | sneqi 3604 |
. . . . . . . 8
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27 | 26 | eqeq2i 2188 |
. . . . . . 7
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28 | 22, 27 | bitr3di 195 |
. . . . . 6
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29 | 15, 28 | bitrid 192 |
. . . . 5
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30 | 12, 29 | mpbid 147 |
. . . 4
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31 | 30 | expcom 116 |
. . 3
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32 | 8, 31 | vtoclga 2803 |
. 2
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33 | 32 | impcom 125 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 4121 ax-pow 4174 ax-pr 4209 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 |
This theorem is referenced by: fressnfv 5703 fnsnsplitss 5715 fnsnsplitdc 6505 dif1en 6878 fnfi 6935 fseq1p1m1 10093 resunimafz0 10810 |
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