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Theorem ffdm 5960
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 5949 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
21feq2d 5929 . . 3 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵𝐹:𝐴𝐵))
32ibir 255 . 2 (𝐹:𝐴𝐵𝐹:dom 𝐹𝐵)
4 eqimss 3619 . . 3 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
51, 4syl 17 . 2 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
63, 5jca 552 1 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wss 3539  dom cdm 5027  wf 5785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-in 3546  df-ss 3553  df-fn 5792  df-f 5793
This theorem is referenced by:  ffdmd  5961  smoiso  7323  s4f1o  13461  islindf2  19919  f1lindf  19927  dfac21  36437  itgperiod  38656  fourierdlem92  38874  fouriersw  38907  etransclem2  38912
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