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Mirrors > Home > MPE Home > Th. List > fsetdmprc0 | Structured version Visualization version GIF version |
Description: The set of functions with a proper class as domain is empty. (Contributed by AV, 22-Aug-2024.) |
Ref | Expression |
---|---|
fsetdmprc0 | ⊢ (𝐴 ∉ V → {𝑓 ∣ 𝑓 Fn 𝐴} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3037 | . . . 4 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V) | |
2 | vex 3467 | . . . . . . 7 ⊢ 𝑔 ∈ V | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (𝑔 Fn 𝐴 → 𝑔 ∈ V) |
4 | id 22 | . . . . . 6 ⊢ (𝑔 Fn 𝐴 → 𝑔 Fn 𝐴) | |
5 | 3, 4 | fndmexd 7910 | . . . . 5 ⊢ (𝑔 Fn 𝐴 → 𝐴 ∈ V) |
6 | 5 | con3i 154 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ 𝑔 Fn 𝐴) |
7 | 1, 6 | sylbi 216 | . . 3 ⊢ (𝐴 ∉ V → ¬ 𝑔 Fn 𝐴) |
8 | 7 | alrimiv 1922 | . 2 ⊢ (𝐴 ∉ V → ∀𝑔 ¬ 𝑔 Fn 𝐴) |
9 | fneq1 6640 | . . 3 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐴 ↔ 𝑔 Fn 𝐴)) | |
10 | 9 | ab0w 4369 | . 2 ⊢ ({𝑓 ∣ 𝑓 Fn 𝐴} = ∅ ↔ ∀𝑔 ¬ 𝑔 Fn 𝐴) |
11 | 8, 10 | sylibr 233 | 1 ⊢ (𝐴 ∉ V → {𝑓 ∣ 𝑓 Fn 𝐴} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1531 = wceq 1533 ∈ wcel 2098 {cab 2702 ∉ wnel 3036 Vcvv 3463 ∅c0 4318 Fn wfn 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nel 3037 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-fun 6545 df-fn 6546 |
This theorem is referenced by: fsetexb 8881 |
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