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Theorem ffdm 5388
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 5373 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
21feq2d 5355 . . 3 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵𝐹:𝐴𝐵))
32ibir 177 . 2 (𝐹:𝐴𝐵𝐹:dom 𝐹𝐵)
4 eqimss 3211 . . 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 1353  wss 3131  dom cdm 4628  wf 5214
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144  df-fn 5221  df-f 5222
This theorem is referenced by:  smoiso  6306  dvcj  14361  dvfre  14362
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