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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 5399. (Contributed by Jim Kingdon, 8-Jan-2019.) |
Ref | Expression |
---|---|
fconstfvm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5066 | . . 3 | |
2 | fvconst 5372 | . . . 4 | |
3 | 2 | ralrimiva 2434 | . . 3 |
4 | 1, 3 | jca 300 | . 2 |
5 | fvelrnb 5242 | . . . . . . . . 9 | |
6 | fveq2 5198 | . . . . . . . . . . . . . 14 | |
7 | 6 | eqeq1d 2089 | . . . . . . . . . . . . 13 |
8 | 7 | rspccva 2700 | . . . . . . . . . . . 12 |
9 | 8 | eqeq1d 2089 | . . . . . . . . . . 11 |
10 | 9 | rexbidva 2365 | . . . . . . . . . 10 |
11 | r19.9rmv 3333 | . . . . . . . . . . 11 | |
12 | 11 | bicomd 139 | . . . . . . . . . 10 |
13 | 10, 12 | sylan9bbr 450 | . . . . . . . . 9 |
14 | 5, 13 | sylan9bbr 450 | . . . . . . . 8 |
15 | velsn 3415 | . . . . . . . . 9 | |
16 | eqcom 2083 | . . . . . . . . 9 | |
17 | 15, 16 | bitr2i 183 | . . . . . . . 8 |
18 | 14, 17 | syl6bb 194 | . . . . . . 7 |
19 | 18 | eqrdv 2079 | . . . . . 6 |
20 | 19 | an32s 532 | . . . . 5 |
21 | 20 | exp31 356 | . . . 4 |
22 | 21 | imdistand 435 | . . 3 |
23 | df-fo 4928 | . . . 4 | |
24 | fof 5126 | . . . 4 | |
25 | 23, 24 | sylbir 133 | . . 3 |
26 | 22, 25 | syl6 33 | . 2 |
27 | 4, 26 | impbid2 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 wral 2348 wrex 2349 csn 3398 crn 4364 wfn 4917 wf 4918 wfo 4920 cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fo 4928 df-fv 4930 |
This theorem is referenced by: fconst3m 5401 |
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