Theorem dmmpt 5126

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
dmmpo.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 4821	. 2 |- dom F = ran `' F
2		dfrn4 5091	. 2 |- ran `' F = ( `' F " _V )
3		dmmpo.1	. . 3 |- F = ( x e. A |-> B )
4	3	mptpreima 5124	. 2 |- ( `' F " _V ) = { x e. A | B e. _V }
5	1, 2, 4	3eqtri 2202	1 |- dom F = { x e. A | B e. _V }

Syntax hints:   = wceq 1353   e. wcel 2148   {crab 2459   _Vcvv 2739   |-> cmpt 4066   `'ccnv 4627   dom cdm 4628   ran crn 4629   "cima 4631

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-mpt 4068  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641

This theorem is referenced by:  dmmptss  5127  dmmptg  5128  dmmptd  5348  fvmptssdm  5602  isnumi  7183  dvrecap  14262