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Theorem ffdm 6757
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 6736 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
21feq2d 6713 . . 3 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵𝐹:𝐴𝐵))
32ibir 267 . 2 (𝐹:𝐴𝐵𝐹:dom 𝐹𝐵)
4 eqimss 4037 . . 3 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
51, 4syl 17 . 2 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
63, 5jca 510 1 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wss 3946  dom cdm 5681  wf 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-cleq 2717  df-ss 3963  df-fn 6556  df-f 6557
This theorem is referenced by:  ffdmd  6758  smoiso  8391  s4f1o  14922  islindf2  21804  fourierdlem92  45756  fouriersw  45789  etransclem2  45794
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