Theorem fndmexd 5558

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 5459	. 2 |- ( ph -> dom F = D )
3		fndmexd.1	. . 3 |- ( ph -> F e. V )
4	3	dmexd 5025	. 2 |- ( ph -> dom F e. _V )
5	2, 4	eqeltrrd 2312	1 |- ( ph -> D e. _V )

Syntax hints:   -> wi 4   e. wcel 2205   _Vcvv 2815   dom cdm 4751   Fn wfn 5349

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-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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-cnv 4759  df-dm 4761  df-rn 4762  df-fn 5357

This theorem is referenced by:  fndmexb  5909  psrbagfsupp  14836  psrbaglesupp  14839  psrbaglecl  14841  psrbagcon  14843  psrbagconf1o  14845