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Mirrors > Home > ILE Home > Th. List > fconstfvm | Unicode version |
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5645. (Contributed by Jim Kingdon, 8-Jan-2019.) |
Ref | Expression |
---|---|
fconstfvm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5280 |
. . 3
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2 | fvconst 5616 |
. . . 4
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3 | 2 | ralrimiva 2508 |
. . 3
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4 | 1, 3 | jca 304 |
. 2
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5 | fvelrnb 5477 |
. . . . . . . . 9
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6 | fveq2 5429 |
. . . . . . . . . . . . . 14
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7 | 6 | eqeq1d 2149 |
. . . . . . . . . . . . 13
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8 | 7 | rspccva 2792 |
. . . . . . . . . . . 12
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9 | 8 | eqeq1d 2149 |
. . . . . . . . . . 11
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10 | 9 | rexbidva 2435 |
. . . . . . . . . 10
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11 | r19.9rmv 3459 |
. . . . . . . . . . 11
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12 | 11 | bicomd 140 |
. . . . . . . . . 10
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13 | 10, 12 | sylan9bbr 459 |
. . . . . . . . 9
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14 | 5, 13 | sylan9bbr 459 |
. . . . . . . 8
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15 | velsn 3549 |
. . . . . . . . 9
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16 | eqcom 2142 |
. . . . . . . . 9
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17 | 15, 16 | bitr2i 184 |
. . . . . . . 8
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18 | 14, 17 | syl6bb 195 |
. . . . . . 7
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19 | 18 | eqrdv 2138 |
. . . . . 6
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20 | 19 | an32s 558 |
. . . . 5
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21 | 20 | exp31 362 |
. . . 4
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22 | 21 | imdistand 444 |
. . 3
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23 | df-fo 5137 |
. . . 4
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24 | fof 5353 |
. . . 4
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25 | 23, 24 | sylbir 134 |
. . 3
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26 | 22, 25 | syl6 33 |
. 2
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27 | 4, 26 | impbid2 142 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fo 5137 df-fv 5139 |
This theorem is referenced by: fconst3m 5647 |
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