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Mirrors > Home > ILE Home > Th. List > fndmin | Unicode version |
Description: Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Ref | Expression |
---|---|
fndmin |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffn5im 5557 |
. . . . . 6
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2 | df-mpt 4063 |
. . . . . 6
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3 | 1, 2 | eqtrdi 2226 |
. . . . 5
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4 | dffn5im 5557 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | df-mpt 4063 |
. . . . . 6
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6 | 4, 5 | eqtrdi 2226 |
. . . . 5
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7 | 3, 6 | ineqan12d 3338 |
. . . 4
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8 | inopab 4755 |
. . . 4
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9 | 7, 8 | eqtrdi 2226 |
. . 3
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10 | 9 | dmeqd 4825 |
. 2
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11 | anandi 590 |
. . . . . . . 8
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12 | 11 | exbii 1605 |
. . . . . . 7
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13 | 19.42v 1906 |
. . . . . . 7
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14 | 12, 13 | bitr3i 186 |
. . . . . 6
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15 | funfvex 5528 |
. . . . . . . . 9
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16 | eqeq1 2184 |
. . . . . . . . . 10
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17 | 16 | ceqsexgv 2866 |
. . . . . . . . 9
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18 | 15, 17 | syl 14 |
. . . . . . . 8
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19 | 18 | funfni 5312 |
. . . . . . 7
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20 | 19 | pm5.32da 452 |
. . . . . 6
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21 | 14, 20 | bitrid 192 |
. . . . 5
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22 | 21 | abbidv 2295 |
. . . 4
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23 | dmopab 4834 |
. . . 4
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24 | df-rab 2464 |
. . . 4
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25 | 22, 23, 24 | 3eqtr4g 2235 |
. . 3
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26 | 25 | adantr 276 |
. 2
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27 | 10, 26 | eqtrd 2210 |
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 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-rab 2464 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-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 |
This theorem is referenced by: fneqeql 5620 mhmeql 12763 |
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