Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > funprg | Unicode version |
Description: A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) |
Ref | Expression |
---|---|
funprg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1010 | . . . 4 | |
2 | simp2l 1012 | . . . 4 | |
3 | funsng 5228 | . . . 4 | |
4 | 1, 2, 3 | syl2anc 409 | . . 3 |
5 | simp1r 1011 | . . . 4 | |
6 | simp2r 1013 | . . . 4 | |
7 | funsng 5228 | . . . 4 | |
8 | 5, 6, 7 | syl2anc 409 | . . 3 |
9 | dmsnopg 5069 | . . . . . 6 | |
10 | 2, 9 | syl 14 | . . . . 5 |
11 | dmsnopg 5069 | . . . . . 6 | |
12 | 6, 11 | syl 14 | . . . . 5 |
13 | 10, 12 | ineq12d 3319 | . . . 4 |
14 | disjsn2 3633 | . . . . 5 | |
15 | 14 | 3ad2ant3 1009 | . . . 4 |
16 | 13, 15 | eqtrd 2197 | . . 3 |
17 | funun 5226 | . . 3 | |
18 | 4, 8, 16, 17 | syl21anc 1226 | . 2 |
19 | df-pr 3577 | . . 3 | |
20 | 19 | funeqi 5203 | . 2 |
21 | 18, 20 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wcel 2135 wne 2334 cun 3109 cin 3110 c0 3404 csn 3570 cpr 3571 cop 3573 cdm 4598 wfun 5176 |
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 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 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 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-fun 5184 |
This theorem is referenced by: funtpg 5233 funpr 5234 fnprg 5237 2strbasg 12438 2stropg 12439 |
Copyright terms: Public domain | W3C validator |