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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 5321. (Contributed by Jim Kingdon, 8-Jan-2019.) |
Ref | Expression |
---|---|
fconstfvm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 4989 |
. . 3
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2 | fvconst 5294 |
. . . 4
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3 | 2 | ralrimiva 2386 |
. . 3
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4 | 1, 3 | jca 290 |
. 2
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5 | fvelrnb 5164 |
. . . . . . . . 9
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6 | fveq2 5121 |
. . . . . . . . . . . . . 14
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7 | 6 | eqeq1d 2045 |
. . . . . . . . . . . . 13
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8 | 7 | rspccva 2649 |
. . . . . . . . . . . 12
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9 | 8 | eqeq1d 2045 |
. . . . . . . . . . 11
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10 | 9 | rexbidva 2317 |
. . . . . . . . . 10
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11 | r19.9rmv 3307 |
. . . . . . . . . . 11
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12 | 11 | bicomd 129 |
. . . . . . . . . 10
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13 | 10, 12 | sylan9bbr 436 |
. . . . . . . . 9
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14 | 5, 13 | sylan9bbr 436 |
. . . . . . . 8
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15 | elsn 3382 |
. . . . . . . . 9
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16 | eqcom 2039 |
. . . . . . . . 9
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17 | 15, 16 | bitr2i 174 |
. . . . . . . 8
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18 | 14, 17 | syl6bb 185 |
. . . . . . 7
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19 | 18 | eqrdv 2035 |
. . . . . 6
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20 | 19 | an32s 502 |
. . . . 5
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21 | 20 | exp31 346 |
. . . 4
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22 | 21 | imdistand 421 |
. . 3
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23 | df-fo 4851 |
. . . 4
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24 | fof 5049 |
. . . 4
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25 | 23, 24 | sylbir 125 |
. . 3
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26 | 22, 25 | syl6 29 |
. 2
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27 | 4, 26 | impbid2 131 |
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 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fo 4851 df-fv 4853 |
This theorem is referenced by: fconst3m 5323 |
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