Theorem ffdmd 6063

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

Hypothesis

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

Assertion

Ref	Expression
ffdmd	|- ( ph -> F : dom F --> B )

Proof of Theorem ffdmd

Step	Hyp	Ref	Expression
1		ffdmd.1	. . 3 |- ( ph -> F : A --> B )
2		ffdm 6062	. . 3 |- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) )
3	1, 2	syl 17	. 2 |- ( ph -> ( F : dom F --> B /\ dom F C_ A ) )
4	3	simpld 475	1 |- ( ph -> F : dom F --> B )

Syntax hints:   -> wi 4   /\ wa 384   C_ wss 3574   dom cdm 5114   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-fn 5891  df-f 5892

This theorem is referenced by:  upgrres1  26205  umgr2v2e  26421  pliguhgr  27338  xlimmnfvlem1  40058  xlimpnfvlem1  40062  issmfd  40944  issmfdf  40946  cnfsmf  40949  issmfled  40966  issmfgtd  40969  smfsuplem1  41017