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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 5234 | . 2 | |
2 | dmsnopg 5070 | . . . . . . . . . 10 | |
3 | 2 | 3ad2ant1 1007 | . . . . . . . . 9 |
4 | dmsnopg 5070 | . . . . . . . . . 10 | |
5 | 4 | 3ad2ant2 1008 | . . . . . . . . 9 |
6 | 3, 5 | jca 304 | . . . . . . . 8 |
7 | 6 | 3ad2ant2 1008 | . . . . . . 7 |
8 | uneq12 3267 | . . . . . . 7 | |
9 | 7, 8 | syl 14 | . . . . . 6 |
10 | df-pr 3578 | . . . . . 6 | |
11 | 9, 10 | eqtr4di 2215 | . . . . 5 |
12 | df-pr 3578 | . . . . . . . 8 | |
13 | 12 | dmeqi 4800 | . . . . . . 7 |
14 | 13 | eqeq1i 2172 | . . . . . 6 |
15 | dmun 4806 | . . . . . . 7 | |
16 | 15 | eqeq1i 2172 | . . . . . 6 |
17 | 14, 16 | bitri 183 | . . . . 5 |
18 | 11, 17 | sylibr 133 | . . . 4 |
19 | dmsnopg 5070 | . . . . . 6 | |
20 | 19 | 3ad2ant3 1009 | . . . . 5 |
21 | 20 | 3ad2ant2 1008 | . . . 4 |
22 | 18, 21 | uneq12d 3273 | . . 3 |
23 | df-tp 3579 | . . . . 5 | |
24 | 23 | dmeqi 4800 | . . . 4 |
25 | dmun 4806 | . . . 4 | |
26 | 24, 25 | eqtri 2185 | . . 3 |
27 | df-tp 3579 | . . 3 | |
28 | 22, 26, 27 | 3eqtr4g 2222 | . 2 |
29 | df-fn 5186 | . 2 | |
30 | 1, 28, 29 | sylanbrc 414 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wcel 2135 wne 2334 cun 3110 csn 3571 cpr 3572 ctp 3573 cop 3574 cdm 4599 wfun 5177 wfn 5178 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2724 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-tp 3579 df-op 3580 df-br 3978 df-opab 4039 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-fun 5185 df-fn 5186 |
This theorem is referenced by: (None) |
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