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Theorem funres11 6405
 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 5847 . 2 (𝐹𝐴) ⊆ 𝐹
2 cnvss 5711 . 2 ((𝐹𝐴) ⊆ 𝐹(𝐹𝐴) ⊆ 𝐹)
3 funss 6347 . 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 3884  ◡ccnv 5522   ↾ cres 5525  Fun wfun 6322 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901  df-br 5034  df-opab 5096  df-rel 5530  df-cnv 5531  df-co 5532  df-res 5535  df-fun 6330 This theorem is referenced by:  f1ssres  6561  resdif  6614  f1ssf1  6625  ssdomg  8542  sbthlem8  8622  spthispth  27518
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