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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 978 | . . . 4 | |
2 | 3simpa 978 | . . . 4 | |
3 | simp1 981 | . . . 4 | |
4 | fprg 5603 | . . . 4 | |
5 | 1, 2, 3, 4 | syl3an 1258 | . . 3 |
6 | eqidd 2140 | . . . 4 | |
7 | simp3 983 | . . . . . . 7 | |
8 | simp3 983 | . . . . . . 7 | |
9 | 7, 8 | anim12i 336 | . . . . . 6 |
10 | 9 | 3adant3 1001 | . . . . 5 |
11 | fsng 5593 | . . . . 5 | |
12 | 10, 11 | syl 14 | . . . 4 |
13 | 6, 12 | mpbird 166 | . . 3 |
14 | df-ne 2309 | . . . . . . 7 | |
15 | df-ne 2309 | . . . . . . 7 | |
16 | elpri 3550 | . . . . . . . . . 10 | |
17 | eqcom 2141 | . . . . . . . . . . 11 | |
18 | eqcom 2141 | . . . . . . . . . . 11 | |
19 | 17, 18 | orbi12i 753 | . . . . . . . . . 10 |
20 | 16, 19 | sylib 121 | . . . . . . . . 9 |
21 | oranim 770 | . . . . . . . . 9 | |
22 | 20, 21 | syl 14 | . . . . . . . 8 |
23 | 22 | con2i 616 | . . . . . . 7 |
24 | 14, 15, 23 | syl2anb 289 | . . . . . 6 |
25 | 24 | 3adant1 999 | . . . . 5 |
26 | 25 | 3ad2ant3 1004 | . . . 4 |
27 | disjsn 3585 | . . . 4 | |
28 | 26, 27 | sylibr 133 | . . 3 |
29 | fun 5295 | . . 3 | |
30 | 5, 13, 28, 29 | syl21anc 1215 | . 2 |
31 | df-tp 3535 | . . . 4 | |
32 | 31 | feq1i 5265 | . . 3 |
33 | df-tp 3535 | . . . 4 | |
34 | df-tp 3535 | . . . 4 | |
35 | 33, 34 | feq23i 5267 | . . 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 697 w3a 962 wceq 1331 wcel 1480 wne 2308 cun 3069 cin 3070 c0 3363 csn 3527 cpr 3528 ctp 3529 cop 3530 wf 5119 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-tp 3535 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 |
This theorem is referenced by: ftp 5605 |
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