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Theorem funres11 5880
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 5342 . 2 (𝐹𝐴) ⊆ 𝐹
2 cnvss 5216 . 2 ((𝐹𝐴) ⊆ 𝐹(𝐹𝐴) ⊆ 𝐹)
3 funss 5822 . 2 ((𝐹𝐴) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹𝐴)))
41, 2, 3mp2b 10 1 (Fun 𝐹 → Fun (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3540  ccnv 5037  cres 5040  Fun wfun 5798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-res 5050  df-fun 5806
This theorem is referenced by:  f1ssres  6021  resdif  6070  ssdomg  7887  sbthlem8  7962  spthispth  26103  f1ssf1  40328  sPthisPth  40932
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