Theorem mapprc 6885

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 e. _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 3533	. . . 4 |- ( E. g g e. { f | f : A --> B } <-> E. f f : A --> B )
2		fdm 5513	. . . . . 6 |- ( f : A --> B -> dom f = A )
3		vex 2815	. . . . . . 7 |- f e. _V
4	3	dmex 5023	. . . . . 6 |- dom f e. _V
5	2, 4	eqeltrrdi 2324	. . . . 5 |- ( f : A --> B -> A e. _V )
6	5	exlimiv 1647	. . . 4 |- ( E. f f : A --> B -> A e. _V )
7	1, 6	sylbi 121	. . 3 |- ( E. g g e. { f | f : A --> B } -> A e. _V )
8	7	con3i 637	. 2 |- ( -. A e. _V -> -. E. g g e. { f | f : A --> B } )
9		notm0 3528	. 2 |- ( -. E. g g e. { f | f : A --> B } <-> { f | f : A --> B } = (/) )
10	8, 9	sylib 122	1 |- ( -. A e. _V -> { f | f : A --> B } = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   E.wex 1541   e. wcel 2203   {cab 2218   _Vcvv 2812   (/)c0 3507   dom cdm 4748   -->wf 5347

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-cnv 4756  df-dm 4758  df-rn 4759  df-fn 5354  df-f 5355

This theorem is referenced by: (None)