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Theorem ffdm 4986
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 4976 . . . 4 (𝐹:AB → dom 𝐹 = A)
21feq2d 4961 . . 3 (𝐹:AB → (𝐹:dom 𝐹B𝐹:AB))
32ibir 166 . 2 (𝐹:AB𝐹:dom 𝐹B)
4 eqimss 2974 . . 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 1228  wss 2894  dom cdm 4272  wf 4825
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-in 2901  df-ss 2908  df-fn 4832  df-f 4833
This theorem is referenced by:  smoiso  5839
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