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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 5725 | . 2 ⊢ ((𝐵 “ {𝐴}) = ∅ ↔ (dom 𝐵 ∩ {𝐴}) = ∅) | |
2 | disjsn 4465 | . 2 ⊢ ((dom 𝐵 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐵) | |
3 | 1, 2 | sylbbr 228 | 1 ⊢ (¬ 𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1658 ∈ wcel 2166 ∩ cin 3797 ∅c0 4144 {csn 4397 dom cdm 5342 “ cima 5345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-xp 5348 df-cnv 5350 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 |
This theorem is referenced by: funfv 6512 dffv2 6518 fpwwe2lem13 9779 bj-funun 33679 |
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