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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 4333 | . . 3 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ≠ ∅ ↔ ∃𝑓 𝑓:𝐴⟶𝐵) | |
2 | fdm 6515 | . . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
3 | vex 3495 | . . . . . 6 ⊢ 𝑓 ∈ V | |
4 | 3 | dmex 7605 | . . . . 5 ⊢ dom 𝑓 ∈ V |
5 | 2, 4 | syl6eqelr 2919 | . . . 4 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
6 | 5 | exlimiv 1922 | . . 3 ⊢ (∃𝑓 𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
7 | 1, 6 | sylbi 218 | . 2 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ≠ ∅ → 𝐴 ∈ V) |
8 | 7 | necon1bi 3041 | 1 ⊢ (¬ 𝐴 ∈ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∃wex 1771 ∈ wcel 2105 {cab 2796 ≠ wne 3013 Vcvv 3492 ∅c0 4288 dom cdm 5548 ⟶wf 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-cnv 5556 df-dm 5558 df-rn 5559 df-fn 6351 df-f 6352 |
This theorem is referenced by: (None) |
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