Theorem fsetdmprc0 6940

Description: The set of functions with a proper class as domain is empty. (Contributed by AV, 22-Aug-2024.)

Assertion

Ref	Expression
fsetdmprc0	|- ( A e/ _V -> { f | f Fn A } = (/) )

Distinct variable group:   A, f

Proof of Theorem fsetdmprc0

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nel 2516	. . . 4 |- ( A e/ _V <-> -. A e. _V )
2		vex 2824	. . . . . . 7 |- g e. _V
3	2	a1i 9	. . . . . 6 |- ( g Fn A -> g e. _V )
4		id 19	. . . . . 6 |- ( g Fn A -> g Fn A )
5	3, 4	fndmexd 5576	. . . . 5 |- ( g Fn A -> A e. _V )
6	5	con3i 641	. . . 4 |- ( -. A e. _V -> -. g Fn A )
7	1, 6	sylbi 121	. . 3 |- ( A e/ _V -> -. g Fn A )
8	7	alrimiv 1927	. 2 |- ( A e/ _V -> A. g -. g Fn A )
9		fneq1 5464	. . 3 |- ( f = g -> ( f Fn A <-> g Fn A ) )
10	9	ab0w 3550	. 2 |- ( { f | f Fn A } = (/) <-> A. g -. g Fn A )
11	8, 10	sylibr 134	1 |- ( A e/ _V -> { f | f Fn A } = (/) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1400   = wceq 1402   e. wcel 2209   {cab 2224   e/ wnel 2515   _Vcvv 2821   (/)c0 3520   Fn wfn 5367

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-nel 2516  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375

This theorem is referenced by: (None)