Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > f0dom0 | Unicode version |
Description: A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.) |
Ref | Expression |
---|---|
f0dom0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq2 5331 | . . . 4 | |
2 | f0bi 5390 | . . . . 5 | |
3 | 2 | biimpi 119 | . . . 4 |
4 | 1, 3 | syl6bi 162 | . . 3 |
5 | 4 | com12 30 | . 2 |
6 | feq1 5330 | . . . 4 | |
7 | fdm 5353 | . . . . 5 | |
8 | dm0 4825 | . . . . 5 | |
9 | 7, 8 | eqtr3di 2218 | . . . 4 |
10 | 6, 9 | syl6bi 162 | . . 3 |
11 | 10 | com12 30 | . 2 |
12 | 5, 11 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1348 c0 3414 cdm 4611 wf 5194 |
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 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 |
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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-fun 5200 df-fn 5201 df-f 5202 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |