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Mirrors > Home > MPE Home > Th. List > ndmima | Structured version Visualization version GIF version |
Description: The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.) (Proof shortened by OpenAI, 3-Jul-2020.) |
Ref | Expression |
---|---|
ndmima | ⊢ (¬ 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imadisj 5941 | . 2 ⊢ ((𝐵 “ {𝐴}) = ∅ ↔ (dom 𝐵 ∩ {𝐴}) = ∅) | |
2 | disjsn 4639 | . 2 ⊢ ((dom 𝐵 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐵) | |
3 | 1, 2 | sylbbr 237 | 1 ⊢ (¬ 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ∅c0 4288 {csn 4557 dom cdm 5548 “ cima 5551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 |
This theorem is referenced by: funfv 6743 dffv2 6749 fpwwe2lem13 10052 bj-funun 34426 |
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