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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvsingle | Structured version Visualization version Unicode version |
Description: The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.) |
Ref | Expression |
---|---|
fvsingle |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5887 |
. . . 4
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2 | sneq 3989 |
. . . 4
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3 | 1, 2 | eqeq12d 2476 |
. . 3
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4 | eqid 2461 |
. . . . 5
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5 | vex 3059 |
. . . . . 6
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6 | snex 4654 |
. . . . . 6
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7 | 5, 6 | brsingle 30732 |
. . . . 5
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8 | 4, 7 | mpbir 214 |
. . . 4
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9 | fnsingle 30734 |
. . . . 5
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10 | fnbrfvb 5927 |
. . . . 5
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11 | 9, 5, 10 | mp2an 683 |
. . . 4
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12 | 8, 11 | mpbir 214 |
. . 3
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13 | 3, 12 | vtoclg 3118 |
. 2
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14 | fvprc 5881 |
. . 3
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15 | snprc 4047 |
. . . 4
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16 | 15 | biimpi 199 |
. . 3
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17 | 14, 16 | eqtr4d 2498 |
. 2
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18 | 13, 17 | pm2.61i 169 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-un 6609 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-ral 2753 df-rex 2754 df-rab 2757 df-v 3058 df-sbc 3279 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-symdif 3674 df-nul 3743 df-if 3893 df-sn 3980 df-pr 3982 df-op 3986 df-uni 4212 df-br 4416 df-opab 4475 df-mpt 4476 df-eprel 4763 df-id 4767 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-iota 5564 df-fun 5602 df-fn 5603 df-f 5604 df-fo 5606 df-fv 5608 df-1st 6819 df-2nd 6820 df-txp 30668 df-singleton 30676 |
This theorem is referenced by: (None) |
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