Theorem dmmpt 5630

Description: The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)

Hypothesis

Ref	Expression
dmmpt.1	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
dmmpt	|- dom F = { x e. A | B e. _V }

Proof of Theorem dmmpt

Step	Hyp	Ref	Expression
1		dfdm4 5316	. 2 |- dom F = ran `' F
2		dfrn4 5595	. 2 |- ran `' F = ( `' F " _V )
3		dmmpt.1	. . 3 |- F = ( x e. A |-> B )
4	3	mptpreima 5628	. 2 |- ( `' F " _V ) = { x e. A | B e. _V }
5	1, 2, 4	3eqtri 2648	1 |- dom F = { x e. A | B e. _V }

Syntax hints:   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  dmmptss  5631  dmmptg  5632  dmmptd  6024  fvmpti  6281  fvmptss  6292  fvmptss2  6304  mptexgf  6485  tz9.12lem3  8652  cardf2  8769  pmtrsn  17939  00lsp  18981  rgrx0ndm  26489  abrexexd  29347  funcnvmptOLD  29467  funcnvmpt  29468  mptctf  29495  issibf  30395  rdgprc0  31699  imageval  32037  dmmptdf  39417  dmmptssf  39438  dmmptdf2  39439  dvcosre  40126  itgsinexplem1  40169  stirlinglem14  40304