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Theorem ffdm 5004
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 4993 . . . 4 (𝐹:AB → dom 𝐹 = A)
21feq2d 4978 . . 3 (𝐹:AB → (𝐹:dom 𝐹B𝐹:AB))
32ibir 166 . 2 (𝐹:AB𝐹:dom 𝐹B)
4 eqimss 2991 . . 3 (dom 𝐹 = A → dom 𝐹A)
51, 4syl 14 . 2 (𝐹:AB → dom 𝐹A)
63, 5jca 290 1 (𝐹:AB → (𝐹:dom 𝐹B dom 𝐹A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wss 2911  dom cdm 4288  wf 4841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-fn 4848  df-f 4849
This theorem is referenced by:  smoiso  5858
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