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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 3047 | . . . 4 ⊢ (𝐴 ∉ V ↔ ¬ 𝐴 ∈ V) | |
2 | vex 3448 | . . . . . . 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 7844 | . . . . 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 1931 | . 2 ⊢ (𝐴 ∉ V → ∀𝑔 ¬ 𝑔 Fn 𝐴) |
9 | fneq1 6594 | . . 3 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐴 ↔ 𝑔 Fn 𝐴)) | |
10 | 9 | ab0w 4334 | . 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 1540 = wceq 1542 ∈ wcel 2107 {cab 2710 ∉ wnel 3046 Vcvv 3444 ∅c0 4283 Fn wfn 6492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nel 3047 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-fun 6499 df-fn 6500 |
This theorem is referenced by: fsetexb 8805 |
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