Theorem ffdm 6732

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 6712	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
2	1	feq2d 6686	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ↔ 𝐹:𝐴⟶𝐵))
3	2	ibir 271	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:dom 𝐹⟶𝐵)
4		eqimss 3989	. . 3 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
5	1, 4	syl 18	. 2 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴)
6	3, 5	jca 521	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ⊆ wss 3899  dom cdm 5655  ⟶wf 6529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2752  df-ss 3916  df-fn 6536  df-f 6537

This theorem is used by:  ffdmd  6733  smoiso  8351  s4f1o  14989  islindf2  22027  fourierdlem92  47026  fouriersw  47059  etransclem2  47064