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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 4376 | . 2 ⊢ (𝐵 “ {𝐴}) = ran (𝐵 ↾ {𝐴}) | |
2 | dmres 4650 | . . . . 5 ⊢ dom (𝐵 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐵) | |
3 | incom 3158 | . . . . 5 ⊢ ({𝐴} ∩ dom 𝐵) = (dom 𝐵 ∩ {𝐴}) | |
4 | 2, 3 | eqtri 2101 | . . . 4 ⊢ dom (𝐵 ↾ {𝐴}) = (dom 𝐵 ∩ {𝐴}) |
5 | disjsn 3454 | . . . . 5 ⊢ ((dom 𝐵 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐵) | |
6 | 5 | biimpri 131 | . . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐵 → (dom 𝐵 ∩ {𝐴}) = ∅) |
7 | 4, 6 | syl5eq 2125 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐵 → dom (𝐵 ↾ {𝐴}) = ∅) |
8 | dm0rn0 4570 | . . 3 ⊢ (dom (𝐵 ↾ {𝐴}) = ∅ ↔ ran (𝐵 ↾ {𝐴}) = ∅) | |
9 | 7, 8 | sylib 120 | . 2 ⊢ (¬ 𝐴 ∈ dom 𝐵 → ran (𝐵 ↾ {𝐴}) = ∅) |
10 | 1, 9 | syl5eq 2125 | 1 ⊢ (¬ 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1284 ∈ wcel 1433 ∩ cin 2972 ∅c0 3251 {csn 3398 dom cdm 4363 ran crn 4364 ↾ cres 4365 “ cima 4366 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 |
This theorem is referenced by: fvun1 5260 |
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