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Theorem ffdm 6277
Description: A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
Assertion
Ref Expression
ffdm (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))

Proof of Theorem ffdm
StepHypRef Expression
1 fdm 6264 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
21feq2d 6242 . . 3 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵𝐹:𝐴𝐵))
32ibir 260 . 2 (𝐹:𝐴𝐵𝐹:dom 𝐹𝐵)
4 eqimss 3853 . . 3 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
51, 4syl 17 . 2 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
63, 5jca 508 1 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wss 3769  dom cdm 5312  wf 6097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-in 3776  df-ss 3783  df-fn 6104  df-f 6105
This theorem is referenced by:  ffdmd  6278  smoiso  7698  s4f1o  14003  islindf2  20478  fourierdlem92  41158  fouriersw  41191  etransclem2  41196
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