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Mirrors > Home > ILE Home > Th. List > ndmima | GIF version |
Description: The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.) |
Ref | Expression |
---|---|
ndmima | ⊢ (¬ 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 4560 | . 2 ⊢ (𝐵 “ {𝐴}) = ran (𝐵 ↾ {𝐴}) | |
2 | dmres 4848 | . . . . 5 ⊢ dom (𝐵 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐵) | |
3 | incom 3273 | . . . . 5 ⊢ ({𝐴} ∩ dom 𝐵) = (dom 𝐵 ∩ {𝐴}) | |
4 | 2, 3 | eqtri 2161 | . . . 4 ⊢ dom (𝐵 ↾ {𝐴}) = (dom 𝐵 ∩ {𝐴}) |
5 | disjsn 3593 | . . . . 5 ⊢ ((dom 𝐵 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐵) | |
6 | 5 | biimpri 132 | . . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐵 → (dom 𝐵 ∩ {𝐴}) = ∅) |
7 | 4, 6 | syl5eq 2185 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐵 → dom (𝐵 ↾ {𝐴}) = ∅) |
8 | dm0rn0 4764 | . . 3 ⊢ (dom (𝐵 ↾ {𝐴}) = ∅ ↔ ran (𝐵 ↾ {𝐴}) = ∅) | |
9 | 7, 8 | sylib 121 | . 2 ⊢ (¬ 𝐴 ∈ dom 𝐵 → ran (𝐵 ↾ {𝐴}) = ∅) |
10 | 1, 9 | syl5eq 2185 | 1 ⊢ (¬ 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1332 ∈ wcel 1481 ∩ cin 3075 ∅c0 3368 {csn 3532 dom cdm 4547 ran crn 4548 ↾ cres 4549 “ cima 4550 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 |
This theorem is referenced by: fvun1 5495 |
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