Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ffdm Structured version   Visualization version   GIF version

Theorem ffdm 6221
 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 6210 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
21feq2d 6190 . . 3 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵𝐹:𝐴𝐵))
32ibir 257 . 2 (𝐹:𝐴𝐵𝐹:dom 𝐹𝐵)
4 eqimss 3796 . . 3 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
51, 4syl 17 . 2 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
63, 5jca 555 1 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1630   ⊆ wss 3713  dom cdm 5264  ⟶wf 6043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-in 3720  df-ss 3727  df-fn 6050  df-f 6051 This theorem is referenced by:  ffdmd  6222  smoiso  7626  s4f1o  13861  islindf2  20353  f1lindf  20361  dfac21  38136  itgperiod  40698  fourierdlem92  40916  fouriersw  40949  etransclem2  40954
 Copyright terms: Public domain W3C validator