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Mirrors > Home > ILE Home > Th. List > dm0rn0 | GIF version |
Description: An empty domain implies an empty range. For a similar theorem for whether the domain and range are inhabited, see dmmrnm 4753. (Contributed by NM, 21-May-1998.) |
Ref | Expression |
---|---|
dm0rn0 | ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alnex 1475 | . . . . . 6 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑥∃𝑦 𝑥𝐴𝑦) | |
2 | excom 1642 | . . . . . 6 ⊢ (∃𝑥∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦∃𝑥 𝑥𝐴𝑦) | |
3 | 1, 2 | xchbinx 671 | . . . . 5 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑦∃𝑥 𝑥𝐴𝑦) |
4 | alnex 1475 | . . . . 5 ⊢ (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ¬ ∃𝑦∃𝑥 𝑥𝐴𝑦) | |
5 | 3, 4 | bitr4i 186 | . . . 4 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦) |
6 | noel 3362 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
7 | 6 | nbn 688 | . . . . 5 ⊢ (¬ ∃𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅)) |
8 | 7 | albii 1446 | . . . 4 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅)) |
9 | noel 3362 | . . . . . 6 ⊢ ¬ 𝑦 ∈ ∅ | |
10 | 9 | nbn 688 | . . . . 5 ⊢ (¬ ∃𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅)) |
11 | 10 | albii 1446 | . . . 4 ⊢ (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅)) |
12 | 5, 8, 11 | 3bitr3i 209 | . . 3 ⊢ (∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅) ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅)) |
13 | abeq1 2247 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅)) | |
14 | abeq1 2247 | . . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅ ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅)) | |
15 | 12, 13, 14 | 3bitr4i 211 | . 2 ⊢ ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅) |
16 | df-dm 4544 | . . 3 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
17 | 16 | eqeq1i 2145 | . 2 ⊢ (dom 𝐴 = ∅ ↔ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅) |
18 | dfrn2 4722 | . . 3 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} | |
19 | 18 | eqeq1i 2145 | . 2 ⊢ (ran 𝐴 = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅) |
20 | 15, 17, 19 | 3bitr4i 211 | 1 ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 ∀wal 1329 = wceq 1331 ∃wex 1468 ∈ wcel 1480 {cab 2123 ∅c0 3358 class class class wbr 3924 dom cdm 4534 ran crn 4535 |
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-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-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-cnv 4542 df-dm 4544 df-rn 4545 |
This theorem is referenced by: rn0 4790 relrn0 4796 imadisj 4896 ndmima 4911 f00 5309 f0rn0 5312 2nd0 6036 map0b 6574 |
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