Theorem mapprc 6826

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 3519	. . . 4 |- ( E. g g e. { f | f : A --> B } <-> E. f f : A --> B )
2		fdm 5490	. . . . . 6 |- ( f : A --> B -> dom f = A )
3		vex 2804	. . . . . . 7 |- f e. _V
4	3	dmex 5001	. . . . . 6 |- dom f e. _V
5	2, 4	eqeltrrdi 2322	. . . . 5 |- ( f : A --> B -> A e. _V )
6	5	exlimiv 1646	. . . 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 3514	. 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 1397   E.wex 1540   e. wcel 2201   {cab 2216   _Vcvv 2801   (/)c0 3493   dom cdm 4727   -->wf 5324

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rex 2515  df-v 2803  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-cnv 4735  df-dm 4737  df-rn 4738  df-fn 5331  df-f 5332

This theorem is referenced by: (None)