Theorem dmmpti 5317

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 5316	. 2 |- F Fn A
4		fndm 5287	. 2 |- ( F Fn A -> dom F = A )
5	3, 4	ax-mp 5	1 |- dom F = A

Syntax hints:   = wceq 1343   e. wcel 2136   _Vcvv 2726   |-> cmpt 4043   dom cdm 4604   Fn wfn 5183

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-fun 5190  df-fn 5191

This theorem is referenced by:  brtpos2  6219  dmtopon  12661