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Theorem funres11 4885
Description: The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
Assertion
Ref Expression
funres11 (Fun 𝐹 → Fun (𝐹A))

Proof of Theorem funres11
StepHypRef Expression
1 resss 4550 . 2 (𝐹A) ⊆ 𝐹
2 cnvss 4423 . 2 ((𝐹A) ⊆ 𝐹(𝐹A) ⊆ 𝐹)
3 funss 4834 . 2 ((𝐹A) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹A)))
41, 2, 3mp2b 8 1 (Fun 𝐹 → Fun (𝐹A))
Colors of variables: wff set class
Syntax hints:  wi 4  wss 2885  ccnv 4259  cres 4262  Fun wfun 4811
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-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528  df-in 2892  df-ss 2899  df-br 3728  df-opab 3782  df-rel 4267  df-cnv 4268  df-co 4269  df-res 4272  df-fun 4819
This theorem is referenced by:  f1ssres  5012  resdif  5061
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