Theorem ffdmd 6631

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

Hypothesis

Ref	Expression
ffdmd.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Assertion

Ref	Expression
ffdmd	⊢ (𝜑 → 𝐹:dom 𝐹⟶𝐵)

Proof of Theorem ffdmd

Step	Hyp	Ref	Expression
1		ffdmd.1	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		ffdm 6630	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
4	3	simpld 495	1 ⊢ (𝜑 → 𝐹:dom 𝐹⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ⊆ wss 3887  dom cdm 5589  ⟶wf 6429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-fn 6436  df-f 6437

This theorem is referenced by:  ordtypelem5  9281  ablfaclem2  19689  ablfac2  19692  f1lindf  21029  lmcnp  22455  upgr1e  27483  upgrres1  27680  umgrres1  27681  umgr2v2e  27892  pliguhgr  28848  s3f1  31221  ccatf1  31223  swrdf1  31228  tocyccntz  31411  dfac21  40891  xlimmnfvlem1  43373  xlimpnfvlem1  43377  itgperiod  43522  issmfd  44271  issmfdf  44273  cnfsmf  44276  issmfled  44293  issmfgtd  44296  smfsuplem1  44344