Theorem dmmpti 5383

Description: Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
fnmpti.1	|- B e. _V
fnmpti.2	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
dmmpti	|- dom F = A

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   F( x)

Proof of Theorem dmmpti

Step	Hyp	Ref	Expression
1		fnmpti.1	. . 3 |- B e. _V
2		fnmpti.2	. . 3 |- F = ( x e. A |-> B )
3	1, 2	fnmpti 5382	. 2 |- F Fn A
4		fndm 5353	. 2 |- ( F Fn A -> dom F = A )
5	3, 4	ax-mp 5	1 |- dom F = A

Syntax hints:   = wceq 1364   e. wcel 2164   _Vcvv 2760   |-> cmpt 4090   dom cdm 4659   Fn wfn 5249

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-fun 5256  df-fn 5257

This theorem is referenced by:  brtpos2  6304  dmtopon  14191