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Mirrors > Home > ILE Home > Th. List > ftpg | Unicode version |
Description: A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
Ref | Expression |
---|---|
ftpg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 979 | . . . 4 | |
2 | 3simpa 979 | . . . 4 | |
3 | simp1 982 | . . . 4 | |
4 | fprg 5650 | . . . 4 | |
5 | 1, 2, 3, 4 | syl3an 1262 | . . 3 |
6 | eqidd 2158 | . . . 4 | |
7 | simp3 984 | . . . . . . 7 | |
8 | simp3 984 | . . . . . . 7 | |
9 | 7, 8 | anim12i 336 | . . . . . 6 |
10 | 9 | 3adant3 1002 | . . . . 5 |
11 | fsng 5640 | . . . . 5 | |
12 | 10, 11 | syl 14 | . . . 4 |
13 | 6, 12 | mpbird 166 | . . 3 |
14 | df-ne 2328 | . . . . . . 7 | |
15 | df-ne 2328 | . . . . . . 7 | |
16 | elpri 3583 | . . . . . . . . . 10 | |
17 | eqcom 2159 | . . . . . . . . . . 11 | |
18 | eqcom 2159 | . . . . . . . . . . 11 | |
19 | 17, 18 | orbi12i 754 | . . . . . . . . . 10 |
20 | 16, 19 | sylib 121 | . . . . . . . . 9 |
21 | oranim 771 | . . . . . . . . 9 | |
22 | 20, 21 | syl 14 | . . . . . . . 8 |
23 | 22 | con2i 617 | . . . . . . 7 |
24 | 14, 15, 23 | syl2anb 289 | . . . . . 6 |
25 | 24 | 3adant1 1000 | . . . . 5 |
26 | 25 | 3ad2ant3 1005 | . . . 4 |
27 | disjsn 3621 | . . . 4 | |
28 | 26, 27 | sylibr 133 | . . 3 |
29 | fun 5342 | . . 3 | |
30 | 5, 13, 28, 29 | syl21anc 1219 | . 2 |
31 | df-tp 3568 | . . . 4 | |
32 | 31 | feq1i 5312 | . . 3 |
33 | df-tp 3568 | . . . 4 | |
34 | df-tp 3568 | . . . 4 | |
35 | 33, 34 | feq23i 5314 | . . 3 |
36 | 32, 35 | bitri 183 | . 2 |
37 | 30, 36 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 w3a 963 wceq 1335 wcel 2128 wne 2327 cun 3100 cin 3101 c0 3394 csn 3560 cpr 3561 ctp 3562 cop 3563 wf 5166 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-tp 3568 df-op 3569 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 |
This theorem is referenced by: ftp 5652 |
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