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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 5637. (Contributed by Jim Kingdon, 8-Jan-2019.) |
Ref | Expression |
---|---|
fconstfvm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5272 | . . 3 | |
2 | fvconst 5608 | . . . 4 | |
3 | 2 | ralrimiva 2505 | . . 3 |
4 | 1, 3 | jca 304 | . 2 |
5 | fvelrnb 5469 | . . . . . . . . 9 | |
6 | fveq2 5421 | . . . . . . . . . . . . . 14 | |
7 | 6 | eqeq1d 2148 | . . . . . . . . . . . . 13 |
8 | 7 | rspccva 2788 | . . . . . . . . . . . 12 |
9 | 8 | eqeq1d 2148 | . . . . . . . . . . 11 |
10 | 9 | rexbidva 2434 | . . . . . . . . . 10 |
11 | r19.9rmv 3454 | . . . . . . . . . . 11 | |
12 | 11 | bicomd 140 | . . . . . . . . . 10 |
13 | 10, 12 | sylan9bbr 458 | . . . . . . . . 9 |
14 | 5, 13 | sylan9bbr 458 | . . . . . . . 8 |
15 | velsn 3544 | . . . . . . . . 9 | |
16 | eqcom 2141 | . . . . . . . . 9 | |
17 | 15, 16 | bitr2i 184 | . . . . . . . 8 |
18 | 14, 17 | syl6bb 195 | . . . . . . 7 |
19 | 18 | eqrdv 2137 | . . . . . 6 |
20 | 19 | an32s 557 | . . . . 5 |
21 | 20 | exp31 361 | . . . 4 |
22 | 21 | imdistand 443 | . . 3 |
23 | df-fo 5129 | . . . 4 | |
24 | fof 5345 | . . . 4 | |
25 | 23, 24 | sylbir 134 | . . 3 |
26 | 22, 25 | syl6 33 | . 2 |
27 | 4, 26 | impbid2 142 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wral 2416 wrex 2417 csn 3527 crn 4540 wfn 5118 wf 5119 wfo 5121 cfv 5123 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fo 5129 df-fv 5131 |
This theorem is referenced by: fconst3m 5639 |
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