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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 4186 | . . 3 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ≠ ∅ ↔ ∃𝑓 𝑓:𝐴⟶𝐵) | |
2 | fdm 6290 | . . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
3 | vex 3417 | . . . . . 6 ⊢ 𝑓 ∈ V | |
4 | 3 | dmex 7366 | . . . . 5 ⊢ dom 𝑓 ∈ V |
5 | 2, 4 | syl6eqelr 2915 | . . . 4 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
6 | 5 | exlimiv 2029 | . . 3 ⊢ (∃𝑓 𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
7 | 1, 6 | sylbi 209 | . 2 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} ≠ ∅ → 𝐴 ∈ V) |
8 | 7 | necon1bi 3027 | 1 ⊢ (¬ 𝐴 ∈ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1656 ∃wex 1878 ∈ wcel 2164 {cab 2811 ≠ wne 2999 Vcvv 3414 ∅c0 4146 dom cdm 5346 ⟶wf 6123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-cnv 5354 df-dm 5356 df-rn 5357 df-fn 6130 df-f 6131 |
This theorem is referenced by: (None) |
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