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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 5368 | . 2 | |
2 | fn0 5237 | . . . . . 6 | |
3 | 2 | biimpi 119 | . . . . 5 |
4 | 3 | adantr 274 | . . . 4 |
5 | dm0 4748 | . . . . 5 | |
6 | cnveq 4708 | . . . . . . . . . 10 | |
7 | cnv0 4937 | . . . . . . . . . 10 | |
8 | 6, 7 | syl6eq 2186 | . . . . . . . . 9 |
9 | 2, 8 | sylbi 120 | . . . . . . . 8 |
10 | 9 | fneq1d 5208 | . . . . . . 7 |
11 | 10 | biimpa 294 | . . . . . 6 |
12 | fndm 5217 | . . . . . 6 | |
13 | 11, 12 | syl 14 | . . . . 5 |
14 | 5, 13 | syl5reqr 2185 | . . . 4 |
15 | 4, 14 | jca 304 | . . 3 |
16 | 2 | biimpri 132 | . . . . 5 |
17 | 16 | adantr 274 | . . . 4 |
18 | eqid 2137 | . . . . . 6 | |
19 | fn0 5237 | . . . . . 6 | |
20 | 18, 19 | mpbir 145 | . . . . 5 |
21 | 8 | fneq1d 5208 | . . . . . 6 |
22 | fneq2 5207 | . . . . . 6 | |
23 | 21, 22 | sylan9bb 457 | . . . . 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 1331 c0 3358 ccnv 4533 cdm 4534 wfn 5113 wf1o 5117 |
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-nul 4049 ax-pow 4093 ax-pr 4126 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 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-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-rn 4545 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 |
This theorem is referenced by: fo00 5396 f1o0 5397 en0 6682 |
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