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Mirrors > Home > ILE Home > Th. List > fntpg | Unicode version |
Description: Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.) |
Ref | Expression |
---|---|
fntpg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funtpg 4950 | . 2 | |
2 | dmsnopg 4792 | . . . . . . . . . 10 | |
3 | 2 | 3ad2ant1 925 | . . . . . . . . 9 |
4 | dmsnopg 4792 | . . . . . . . . . 10 | |
5 | 4 | 3ad2ant2 926 | . . . . . . . . 9 |
6 | 3, 5 | jca 290 | . . . . . . . 8 |
7 | 6 | 3ad2ant2 926 | . . . . . . 7 |
8 | uneq12 3092 | . . . . . . 7 | |
9 | 7, 8 | syl 14 | . . . . . 6 |
10 | df-pr 3382 | . . . . . 6 | |
11 | 9, 10 | syl6eqr 2090 | . . . . 5 |
12 | df-pr 3382 | . . . . . . . 8 | |
13 | 12 | dmeqi 4536 | . . . . . . 7 |
14 | 13 | eqeq1i 2047 | . . . . . 6 |
15 | dmun 4542 | . . . . . . 7 | |
16 | 15 | eqeq1i 2047 | . . . . . 6 |
17 | 14, 16 | bitri 173 | . . . . 5 |
18 | 11, 17 | sylibr 137 | . . . 4 |
19 | dmsnopg 4792 | . . . . . 6 | |
20 | 19 | 3ad2ant3 927 | . . . . 5 |
21 | 20 | 3ad2ant2 926 | . . . 4 |
22 | 18, 21 | uneq12d 3098 | . . 3 |
23 | df-tp 3383 | . . . . 5 | |
24 | 23 | dmeqi 4536 | . . . 4 |
25 | dmun 4542 | . . . 4 | |
26 | 24, 25 | eqtri 2060 | . . 3 |
27 | df-tp 3383 | . . 3 | |
28 | 22, 26, 27 | 3eqtr4g 2097 | . 2 |
29 | df-fn 4905 | . 2 | |
30 | 1, 28, 29 | sylanbrc 394 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 wne 2204 cun 2915 csn 3375 cpr 3376 ctp 3377 cop 3378 cdm 4345 wfun 4896 wfn 4897 |
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-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-tp 3383 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-fun 4904 df-fn 4905 |
This theorem is referenced by: (None) |
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