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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 4970 | . 2 | |
2 | dmsnopg 4812 | . . . . . . . . . 10 | |
3 | 2 | 3ad2ant1 959 | . . . . . . . . 9 |
4 | dmsnopg 4812 | . . . . . . . . . 10 | |
5 | 4 | 3ad2ant2 960 | . . . . . . . . 9 |
6 | 3, 5 | jca 300 | . . . . . . . 8 |
7 | 6 | 3ad2ant2 960 | . . . . . . 7 |
8 | uneq12 3121 | . . . . . . 7 | |
9 | 7, 8 | syl 14 | . . . . . 6 |
10 | df-pr 3405 | . . . . . 6 | |
11 | 9, 10 | syl6eqr 2131 | . . . . 5 |
12 | df-pr 3405 | . . . . . . . 8 | |
13 | 12 | dmeqi 4554 | . . . . . . 7 |
14 | 13 | eqeq1i 2088 | . . . . . 6 |
15 | dmun 4560 | . . . . . . 7 | |
16 | 15 | eqeq1i 2088 | . . . . . 6 |
17 | 14, 16 | bitri 182 | . . . . 5 |
18 | 11, 17 | sylibr 132 | . . . 4 |
19 | dmsnopg 4812 | . . . . . 6 | |
20 | 19 | 3ad2ant3 961 | . . . . 5 |
21 | 20 | 3ad2ant2 960 | . . . 4 |
22 | 18, 21 | uneq12d 3127 | . . 3 |
23 | df-tp 3406 | . . . . 5 | |
24 | 23 | dmeqi 4554 | . . . 4 |
25 | dmun 4560 | . . . 4 | |
26 | 24, 25 | eqtri 2101 | . . 3 |
27 | df-tp 3406 | . . 3 | |
28 | 22, 26, 27 | 3eqtr4g 2138 | . 2 |
29 | df-fn 4925 | . 2 | |
30 | 1, 28, 29 | sylanbrc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wcel 1433 wne 2245 cun 2971 csn 3398 cpr 3399 ctp 3400 cop 3401 cdm 4363 wfun 4916 wfn 4917 |
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-in1 576 ax-in2 577 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-3or 920 df-3an 921 df-tru 1287 df-fal 1290 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-ne 2246 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-tp 3406 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-fun 4924 df-fn 4925 |
This theorem is referenced by: (None) |
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