Theorem ffdm 6777

Description: A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)

Assertion

Ref	Expression
ffdm	⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))

Proof of Theorem ffdm

Step	Hyp	Ref	Expression
1		fdm 6756	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
2	1	feq2d 6733	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ↔ 𝐹:𝐴⟶𝐵))
3	2	ibir 268	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:dom 𝐹⟶𝐵)
4		eqimss 4067	. . 3 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
5	1, 4	syl 17	. 2 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴)
6	3, 5	jca 511	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ⊆ wss 3976  dom cdm 5700  ⟶wf 6569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732  df-ss 3993  df-fn 6576  df-f 6577

This theorem is referenced by:  ffdmd  6778  smoiso  8418  s4f1o  14967  islindf2  21857  fourierdlem92  46119  fouriersw  46152  etransclem2  46157