Theorem mapprc 3260

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.

Assertion

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

Distinct variable group(s):   A,f   B,f

Proof of Theorem mapprc

Step	Hyp	Ref	Expression
1		abn0 1715	. . 3 ⊢ (¬ {f∣f:A–→B} = ∅ ↔ ∃f f:A–→B)
2		visset 1350	. . . . . 6 ⊢ f ∈ V
3		dmexg 2551	. . . . . 6 ⊢ (f ∈ V → dom f ∈ V)
4	2, 3	ax-mp 6	. . . . 5 ⊢ dom f ∈ V
5		fdm 2756	. . . . . 6 ⊢ (f:A–→B → dom f = A)
6	5	eleq1d 1155	. . . . 5 ⊢ (f:A–→B → (dom f ∈ V ↔ A ∈ V))
7	4, 6	mpbii 168	. . . 4 ⊢ (f:A–→B → A ∈ V)
8	7	19.23aiv 952	. . 3 ⊢ (∃f f:A–→B → A ∈ V)
9	1, 8	sylbi 174	. 2 ⊢ (¬ {f∣f:A–→B} = ∅ → A ∈ V)
10	9	con1i 88	1 ⊢ (¬ A ∈ V → {f∣f:A–→B} = ∅)

Syntax hints:  ¬ wn 1   → wi 2  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  dom cdm 2410  –→wf 2418

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-dm 2428  df-fn 2433  df-f 2434

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