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Mirrors > Home > ILE Home > Th. List > f1o00 | Unicode version |
Description: One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.) |
Ref | Expression |
---|---|
f1o00 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff1o4 5434 | . 2 | |
2 | fn0 5301 | . . . . . 6 | |
3 | 2 | biimpi 119 | . . . . 5 |
4 | 3 | adantr 274 | . . . 4 |
5 | cnveq 4772 | . . . . . . . . . 10 | |
6 | cnv0 5001 | . . . . . . . . . 10 | |
7 | 5, 6 | eqtrdi 2213 | . . . . . . . . 9 |
8 | 2, 7 | sylbi 120 | . . . . . . . 8 |
9 | 8 | fneq1d 5272 | . . . . . . 7 |
10 | 9 | biimpa 294 | . . . . . 6 |
11 | fndm 5281 | . . . . . 6 | |
12 | 10, 11 | syl 14 | . . . . 5 |
13 | dm0 4812 | . . . . 5 | |
14 | 12, 13 | eqtr3di 2212 | . . . 4 |
15 | 4, 14 | jca 304 | . . 3 |
16 | 2 | biimpri 132 | . . . . 5 |
17 | 16 | adantr 274 | . . . 4 |
18 | eqid 2164 | . . . . . 6 | |
19 | fn0 5301 | . . . . . 6 | |
20 | 18, 19 | mpbir 145 | . . . . 5 |
21 | 7 | fneq1d 5272 | . . . . . 6 |
22 | fneq2 5271 | . . . . . 6 | |
23 | 21, 22 | sylan9bb 458 | . . . . 5 |
24 | 20, 23 | mpbiri 167 | . . . 4 |
25 | 17, 24 | jca 304 | . . 3 |
26 | 15, 25 | impbii 125 | . 2 |
27 | 1, 26 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1342 c0 3404 ccnv 4597 cdm 4598 wfn 5177 wf1o 5181 |
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-nul 4102 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-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-rn 4609 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 |
This theorem is referenced by: fo00 5462 f1o0 5463 en0 6752 |
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