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Mirrors > Home > ILE Home > Th. List > f0rn0 | Unicode version |
Description: If there is no element in the range of a function, its domain must be empty. (Contributed by Alexander van der Vekens, 12-Jul-2018.) |
Ref | Expression |
---|---|
f0rn0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdm 5351 | . . 3 | |
2 | frn 5354 | . . . . . . . . 9 | |
3 | ralnex 2458 | . . . . . . . . . 10 | |
4 | disj 3462 | . . . . . . . . . . 11 | |
5 | df-ss 3134 | . . . . . . . . . . . 12 | |
6 | incom 3319 | . . . . . . . . . . . . . 14 | |
7 | 6 | eqeq1i 2178 | . . . . . . . . . . . . 13 |
8 | eqtr2 2189 | . . . . . . . . . . . . . 14 | |
9 | 8 | ex 114 | . . . . . . . . . . . . 13 |
10 | 7, 9 | sylbi 120 | . . . . . . . . . . . 12 |
11 | 5, 10 | sylbi 120 | . . . . . . . . . . 11 |
12 | 4, 11 | syl5bir 152 | . . . . . . . . . 10 |
13 | 3, 12 | syl5bir 152 | . . . . . . . . 9 |
14 | 2, 13 | syl 14 | . . . . . . . 8 |
15 | 14 | imp 123 | . . . . . . 7 |
16 | 15 | adantl 275 | . . . . . 6 |
17 | dm0rn0 4826 | . . . . . 6 | |
18 | 16, 17 | sylibr 133 | . . . . 5 |
19 | eqeq1 2177 | . . . . . . 7 | |
20 | 19 | eqcoms 2173 | . . . . . 6 |
21 | 20 | adantr 274 | . . . . 5 |
22 | 18, 21 | mpbird 166 | . . . 4 |
23 | 22 | exp32 363 | . . 3 |
24 | 1, 23 | mpcom 36 | . 2 |
25 | 24 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 wrex 2449 cin 3120 wss 3121 c0 3414 cdm 4609 crn 4610 wf 5192 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-cnv 4617 df-dm 4619 df-rn 4620 df-fn 5199 df-f 5200 |
This theorem is referenced by: (None) |
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