Theorem fndmexd 5581

Description: If a function is a set, its domain is a set. (Contributed by Rohan Ridenour, 13-May-2024.)

Hypotheses

Ref	Expression
fndmexd.1	|- ( ph -> F e. V )
fndmexd.2	|- ( ph -> F Fn D )

Assertion

Ref	Expression
fndmexd	|- ( ph -> D e. _V )

Proof of Theorem fndmexd

Step	Hyp	Ref	Expression
1		fndmexd.2	. . 3 |- ( ph -> F Fn D )
2	1	fndmd 5482	. 2 |- ( ph -> dom F = D )
3		fndmexd.1	. . 3 |- ( ph -> F e. V )
4	3	dmexd 5048	. 2 |- ( ph -> dom F e. _V )
5	2, 4	eqeltrrd 2316	1 |- ( ph -> D e. _V )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   _Vcvv 2821   dom cdm 4774   Fn wfn 5372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-cnv 4782  df-dm 4784  df-rn 4785  df-fn 5380

This theorem is used by:  fndmexb  5938  fsetdmprc0  6950  psrbagfsupp  15055  psrbaglesupp  15058  psrbaglecl  15060  psrbagcon  15062  psrbagconf1o  15064