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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 4336 | . . 3 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ≠ ∅ ↔ ∃𝑓 𝑓:𝐴⟶𝐵) | |
2 | fdm 6522 | . . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
3 | vex 3497 | . . . . . 6 ⊢ 𝑓 ∈ V | |
4 | 3 | dmex 7616 | . . . . 5 ⊢ dom 𝑓 ∈ V |
5 | 2, 4 | eqeltrrdi 2922 | . . . 4 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
6 | 5 | exlimiv 1931 | . . 3 ⊢ (∃𝑓 𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
7 | 1, 6 | sylbi 219 | . 2 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ≠ ∅ → 𝐴 ∈ V) |
8 | 7 | necon1bi 3044 | 1 ⊢ (¬ 𝐴 ∈ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2799 ≠ wne 3016 Vcvv 3494 ∅c0 4291 dom cdm 5555 ⟶wf 6351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-cnv 5563 df-dm 5565 df-rn 5566 df-fn 6358 df-f 6359 |
This theorem is referenced by: efmndbasabf 18037 |
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