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Mirrors > Home > ILE Home > Th. List > funtpg | Unicode version |
Description: A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.) |
Ref | Expression |
---|---|
funtpg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 978 | . . . 4 | |
2 | 3simpa 978 | . . . 4 | |
3 | simp1 981 | . . . 4 | |
4 | funprg 5168 | . . . 4 | |
5 | 1, 2, 3, 4 | syl3an 1258 | . . 3 |
6 | simp13 1013 | . . . 4 | |
7 | simp23 1016 | . . . 4 | |
8 | funsng 5164 | . . . 4 | |
9 | 6, 7, 8 | syl2anc 408 | . . 3 |
10 | 2 | 3ad2ant2 1003 | . . . . . 6 |
11 | dmpropg 5006 | . . . . . 6 | |
12 | 10, 11 | syl 14 | . . . . 5 |
13 | dmsnopg 5005 | . . . . . 6 | |
14 | 7, 13 | syl 14 | . . . . 5 |
15 | 12, 14 | ineq12d 3273 | . . . 4 |
16 | elpri 3545 | . . . . . . . 8 | |
17 | nner 2310 | . . . . . . . . . . . 12 | |
18 | 17 | eqcoms 2140 | . . . . . . . . . . 11 |
19 | 3mix2 1151 | . . . . . . . . . . 11 | |
20 | 18, 19 | syl 14 | . . . . . . . . . 10 |
21 | nner 2310 | . . . . . . . . . . . 12 | |
22 | 21 | eqcoms 2140 | . . . . . . . . . . 11 |
23 | 3mix3 1152 | . . . . . . . . . . 11 | |
24 | 22, 23 | syl 14 | . . . . . . . . . 10 |
25 | 20, 24 | jaoi 705 | . . . . . . . . 9 |
26 | 3ianorr 1287 | . . . . . . . . 9 | |
27 | 25, 26 | syl 14 | . . . . . . . 8 |
28 | 16, 27 | syl 14 | . . . . . . 7 |
29 | 28 | con2i 616 | . . . . . 6 |
30 | disjsn 3580 | . . . . . 6 | |
31 | 29, 30 | sylibr 133 | . . . . 5 |
32 | 31 | 3ad2ant3 1004 | . . . 4 |
33 | 15, 32 | eqtrd 2170 | . . 3 |
34 | funun 5162 | . . 3 | |
35 | 5, 9, 33, 34 | syl21anc 1215 | . 2 |
36 | df-tp 3530 | . . 3 | |
37 | 36 | funeqi 5139 | . 2 |
38 | 35, 37 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 w3o 961 w3a 962 wceq 1331 wcel 1480 wne 2306 cun 3064 cin 3065 c0 3358 csn 3522 cpr 3523 ctp 3524 cop 3525 cdm 4534 wfun 5112 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-tp 3530 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-fun 5120 |
This theorem is referenced by: fntpg 5174 |
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