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Theorem ffdm 5293
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 5278 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
21feq2d 5260 . . 3 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵𝐹:𝐴𝐵))
32ibir 176 . 2 (𝐹:𝐴𝐵𝐹:dom 𝐹𝐵)
4 eqimss 3151 . . 3 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
51, 4syl 14 . 2 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
63, 5jca 304 1 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wss 3071  dom cdm 4539  wf 5119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084  df-fn 5126  df-f 5127
This theorem is referenced by:  smoiso  6199  dvcj  12856  dvfre  12857
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