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Mirrors > Home > MPE Home > Th. List > mapprc | Structured version Visualization version GIF version |
Description: When 𝐴 is a proper class, the class of all functions mapping 𝐴 to 𝐵 is empty. Exercise 4.41 of [Mendelson] p. 255. (Contributed by NM, 8-Dec-2003.) |
Ref | Expression |
---|---|
mapprc | ⊢ (¬ 𝐴 ∈ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abn0 4381 | . . 3 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ≠ ∅ ↔ ∃𝑓 𝑓:𝐴⟶𝐵) | |
2 | fdm 6731 | . . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
3 | vex 3475 | . . . . . 6 ⊢ 𝑓 ∈ V | |
4 | 3 | dmex 7917 | . . . . 5 ⊢ dom 𝑓 ∈ V |
5 | 2, 4 | eqeltrrdi 2838 | . . . 4 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
6 | 5 | exlimiv 1926 | . . 3 ⊢ (∃𝑓 𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
7 | 1, 6 | sylbi 216 | . 2 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ≠ ∅ → 𝐴 ∈ V) |
8 | 7 | necon1bi 2966 | 1 ⊢ (¬ 𝐴 ∈ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∃wex 1774 ∈ wcel 2099 {cab 2705 ≠ wne 2937 Vcvv 3471 ∅c0 4323 dom cdm 5678 ⟶wf 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-cnv 5686 df-dm 5688 df-rn 5689 df-fn 6551 df-f 6552 |
This theorem is referenced by: efmndbasabf 18824 |
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