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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 4385 | . . 3 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ≠ ∅ ↔ ∃𝑓 𝑓:𝐴⟶𝐵) | |
| 2 | fdm 6745 | . . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
| 3 | vex 3484 | . . . . . 6 ⊢ 𝑓 ∈ V | |
| 4 | 3 | dmex 7931 | . . . . 5 ⊢ dom 𝑓 ∈ V | 
| 5 | 2, 4 | eqeltrrdi 2850 | . . . 4 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) | 
| 6 | 5 | exlimiv 1930 | . . 3 ⊢ (∃𝑓 𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) | 
| 7 | 1, 6 | sylbi 217 | . 2 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ≠ ∅ → 𝐴 ∈ V) | 
| 8 | 7 | necon1bi 2969 | 1 ⊢ (¬ 𝐴 ∈ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ≠ wne 2940 Vcvv 3480 ∅c0 4333 dom cdm 5685 ⟶wf 6557 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-fn 6564 df-f 6565 | 
| This theorem is referenced by: efmndbasabf 18885 | 
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