Theorem fdmd 39420

Description: The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
fdmd.1	|- ( ph -> F : A --> B )

Assertion

Ref	Expression
fdmd	|- ( ph -> dom F = A )

Proof of Theorem fdmd

Step	Hyp	Ref	Expression
1		fdmd.1	. 2 |- ( ph -> F : A --> B )
2		fdm 6051	. 2 |- ( F : A --> B -> dom F = A )
3	1, 2	syl 17	1 |- ( ph -> dom F = A )

Syntax hints:   -> wi 4   = wceq 1483   dom cdm 5114   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386  df-fn 5891  df-f 5892

This theorem is referenced by:  limsuppnfdlem  39933  limsupvaluz  39940  climxrrelem  39981  climxrre  39982  liminfvalxr  40015  xlimmnfvlem2  40059  xlimpnfvlem2  40063  issmfd  40944  issmfdf  40946  cnfsmf  40949  issmfled  40966  smfmbfcex  40968  issmfgtd  40969  smfsuplem1  41017