Theorem mapprc 6799

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.)

Assertion

Ref	Expression
mapprc	⊢ (¬ 𝐴 ∈ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = ∅)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem mapprc

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abn0m 3517	. . . 4 ⊢ (∃𝑔 𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ ∃𝑓 𝑓:𝐴⟶𝐵)
2		fdm 5479	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴)
3		vex 2802	. . . . . . 7 ⊢ 𝑓 ∈ V
4	3	dmex 4991	. . . . . 6 ⊢ dom 𝑓 ∈ V
5	2, 4	eqeltrrdi 2321	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V)
6	5	exlimiv 1644	. . . 4 ⊢ (∃𝑓 𝑓:𝐴⟶𝐵 → 𝐴 ∈ V)
7	1, 6	sylbi 121	. . 3 ⊢ (∃𝑔 𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} → 𝐴 ∈ V)
8	7	con3i 635	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑔 𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵})
9		notm0 3512	. 2 ⊢ (¬ ∃𝑔 𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ {𝑓 ∣ 𝑓:𝐴⟶𝐵} = ∅)
10	8, 9	sylib 122	1 ⊢ (¬ 𝐴 ∈ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215  Vcvv 2799  ∅c0 3491  dom cdm 4719  ⟶wf 5314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-cnv 4727  df-dm 4729  df-rn 4730  df-fn 5321  df-f 5322

This theorem is referenced by: (None)