Theorem ffdm 5235

Description: A mapping is a partial function. (Contributed by set.mm contributors, 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 5227	. . . 4 ⊢ (F:A–→B → dom F = A)
2	1	feq2d 5216	. . 3 ⊢ (F:A–→B → (F:dom F–→B ↔ F:A–→B))
3	2	ibir 233	. 2 ⊢ (F:A–→B → F:dom F–→B)
4		eqimss 3324	. . 3 ⊢ (dom F = A → dom F ⊆ A)
5	1, 4	syl 15	. 2 ⊢ (F:A–→B → dom F ⊆ A)
6	3, 5	jca 518	1 ⊢ (F:A–→B → (F:dom F–→B ∧ dom F ⊆ A))

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ⊆ wss 3258  dom cdm 4773  –→wf 4778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-fn 4791  df-f 4792

This theorem is referenced by: (None)