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Theorem funres11 6642
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 6018 . 2 (𝐹𝐴) ⊆ 𝐹
2 cnvss 5882 . 2 ((𝐹𝐴) ⊆ 𝐹(𝐹𝐴) ⊆ 𝐹)
3 funss 6584 . 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 3950  ccnv 5683  cres 5686  Fun wfun 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-in 3957  df-ss 3967  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-res 5696  df-fun 6562
This theorem is referenced by:  f1ssres  6810  resdif  6868  f1ssf1  6879  resf1extb  7957  ssdomg  9041  sbthlem8  9131  spthispth  29745
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