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Theorem ffdm 5234
 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
StepHypRef Expression
1 fdm 5226 . . . 4 (F:A–→B → dom F = A)
21feq2d 5215 . . 3 (F:A–→B → (F:dom F–→BF:A–→B))
32ibir 233 . 2 (F:A–→BF:dom F–→B)
4 eqimss 3323 . . 3 (dom F = A → dom F A)
51, 4syl 15 . 2 (F:A–→B → dom F A)
63, 5jca 518 1 (F:A–→B → (F:dom F–→B dom F A))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ⊆ wss 3257  dom cdm 4772  –→wf 4777 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-fn 4790  df-f 4791 This theorem is referenced by: (None)
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