Theorem fdmd 39938

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

Hypothesis

Ref	Expression
fdmd.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Assertion

Ref	Expression
fdmd	⊢ (𝜑 → dom 𝐹 = 𝐴)

Proof of Theorem fdmd

Step	Hyp	Ref	Expression
1		fdmd.1	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		fdm 6191	. 2 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
3	1, 2	syl 17	1 ⊢ (𝜑 → dom 𝐹 = 𝐴)

Syntax hints:   → wi 4   = wceq 1631  dom cdm 5249  ⟶wf 6027

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 383  df-fn 6034  df-f 6035

This theorem is referenced by:  limsuppnfdlem  40451  limsupvaluz  40458  climxrrelem  40499  climxrre  40500  liminfvalxr  40533  xlimmnfvlem2  40577  xlimpnfvlem2  40581  issmfd  41464  issmfdf  41466  cnfsmf  41469  issmfled  41486  smfmbfcex  41488  issmfgtd  41489  smfsuplem1  41537