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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 5144 | . 2 | |
2 | dmsnopg 4980 | . . . . . . . . . 10 | |
3 | 2 | 3ad2ant1 987 | . . . . . . . . 9 |
4 | dmsnopg 4980 | . . . . . . . . . 10 | |
5 | 4 | 3ad2ant2 988 | . . . . . . . . 9 |
6 | 3, 5 | jca 304 | . . . . . . . 8 |
7 | 6 | 3ad2ant2 988 | . . . . . . 7 |
8 | uneq12 3195 | . . . . . . 7 | |
9 | 7, 8 | syl 14 | . . . . . 6 |
10 | df-pr 3504 | . . . . . 6 | |
11 | 9, 10 | syl6eqr 2168 | . . . . 5 |
12 | df-pr 3504 | . . . . . . . 8 | |
13 | 12 | dmeqi 4710 | . . . . . . 7 |
14 | 13 | eqeq1i 2125 | . . . . . 6 |
15 | dmun 4716 | . . . . . . 7 | |
16 | 15 | eqeq1i 2125 | . . . . . 6 |
17 | 14, 16 | bitri 183 | . . . . 5 |
18 | 11, 17 | sylibr 133 | . . . 4 |
19 | dmsnopg 4980 | . . . . . 6 | |
20 | 19 | 3ad2ant3 989 | . . . . 5 |
21 | 20 | 3ad2ant2 988 | . . . 4 |
22 | 18, 21 | uneq12d 3201 | . . 3 |
23 | df-tp 3505 | . . . . 5 | |
24 | 23 | dmeqi 4710 | . . . 4 |
25 | dmun 4716 | . . . 4 | |
26 | 24, 25 | eqtri 2138 | . . 3 |
27 | df-tp 3505 | . . 3 | |
28 | 22, 26, 27 | 3eqtr4g 2175 | . 2 |
29 | df-fn 5096 | . 2 | |
30 | 1, 28, 29 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 wne 2285 cun 3039 csn 3497 cpr 3498 ctp 3499 cop 3500 cdm 4509 wfun 5087 wfn 5088 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-tp 3505 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-fun 5095 df-fn 5096 |
This theorem is referenced by: (None) |
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