Theorem dmmpti 5327

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 5326	. 2 |- F Fn A
4		fndm 5297	. 2 |- ( F Fn A -> dom F = A )
5	3, 4	ax-mp 5	1 |- dom F = A

Syntax hints:   = wceq 1348   e. wcel 2141   _Vcvv 2730   |-> cmpt 4050   dom cdm 4611   Fn wfn 5193

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-fun 5200  df-fn 5201

This theorem is referenced by:  brtpos2  6230  dmtopon  12815