Theorem dmmpti 5260

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 5259	. 2 |- F Fn A
4		fndm 5230	. 2 |- ( F Fn A -> dom F = A )
5	3, 4	ax-mp 5	1 |- dom F = A

Syntax hints:   = wceq 1332   e. wcel 1481   _Vcvv 2689   |-> cmpt 3997   dom cdm 4547   Fn wfn 5126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-fun 5133  df-fn 5134

This theorem is referenced by:  brtpos2  6156  dmtopon  12229