Theorem ffdmd 6511

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 6510	. . 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 498	1 ⊢ (𝜑 → 𝐹:dom 𝐹⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ⊆ wss 3881  dom cdm 5519  ⟶wf 6320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-fn 6327  df-f 6328

This theorem is referenced by:  ordtypelem5  8970  ablfaclem2  19201  ablfac2  19204  f1lindf  20511  lmcnp  21909  upgr1e  26906  upgrres1  27103  umgrres1  27104  umgr2v2e  27315  pliguhgr  28269  s3f1  30649  ccatf1  30651  swrdf1  30656  tocyccntz  30836  dfac21  40010  xlimmnfvlem1  42474  xlimpnfvlem1  42478  itgperiod  42623  issmfd  43369  issmfdf  43371  cnfsmf  43374  issmfled  43391  issmfgtd  43394  smfsuplem1  43442