Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > f1osng | Structured version Visualization version Unicode version |
Description: A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.) |
Ref | Expression |
---|---|
f1osng |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4187 | . . . 4 | |
2 | f1oeq2 6128 | . . . 4 | |
3 | 1, 2 | syl 17 | . . 3 |
4 | opeq1 4402 | . . . . 5 | |
5 | 4 | sneqd 4189 | . . . 4 |
6 | f1oeq1 6127 | . . . 4 | |
7 | 5, 6 | syl 17 | . . 3 |
8 | 3, 7 | bitrd 268 | . 2 |
9 | sneq 4187 | . . . 4 | |
10 | f1oeq3 6129 | . . . 4 | |
11 | 9, 10 | syl 17 | . . 3 |
12 | opeq2 4403 | . . . . 5 | |
13 | 12 | sneqd 4189 | . . . 4 |
14 | f1oeq1 6127 | . . . 4 | |
15 | 13, 14 | syl 17 | . . 3 |
16 | 11, 15 | bitrd 268 | . 2 |
17 | vex 3203 | . . 3 | |
18 | vex 3203 | . . 3 | |
19 | 17, 18 | f1osn 6176 | . 2 |
20 | 8, 16, 19 | vtocl2g 3270 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 csn 4177 cop 4183 wf1o 5887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 |
This theorem is referenced by: f1sng 6178 f1oprswap 6180 f1oprg 6181 f1o2sn 6408 fsnunf 6451 fsnex 6538 suppsnop 7309 ralxpmap 7907 enfixsn 8069 fseqenlem1 8847 canthp1lem2 9475 sumsnf 14473 sumsn 14475 prodsn 14692 prodsnf 14694 vdwlem8 15692 gsumws1 17376 symg1bas 17816 dprdsn 18435 eupthp1 27076 poimirlem16 33425 poimirlem17 33426 poimirlem19 33428 poimirlem20 33429 mapfzcons 37279 sumsnd 39185 mapsnd 39388 1hegrlfgr 41713 |
Copyright terms: Public domain | W3C validator |