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Theorem ffdm 5398
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 5383 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
21feq2d 5365 . . 3 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵𝐹:𝐴𝐵))
32ibir 177 . 2 (𝐹:𝐴𝐵𝐹:dom 𝐹𝐵)
4 eqimss 3221 . . 3 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
51, 4syl 14 . 2 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
63, 5jca 306 1 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  wss 3141  dom cdm 4638  wf 5224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-in 3147  df-ss 3154  df-fn 5231  df-f 5232
This theorem is referenced by:  smoiso  6317  dvcj  14526  dvfre  14527
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