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Theorem funres11 6579
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 5963 . 2 (𝐹𝐴) ⊆ 𝐹
2 cnvss 5829 . 2 ((𝐹𝐴) ⊆ 𝐹(𝐹𝐴) ⊆ 𝐹)
3 funss 6521 . 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 3911  ccnv 5633  cres 5636  Fun wfun 6491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-in 3918  df-ss 3928  df-br 5107  df-opab 5169  df-rel 5641  df-cnv 5642  df-co 5643  df-res 5646  df-fun 6499
This theorem is referenced by:  f1ssres  6747  resdif  6806  f1ssf1  6817  ssdomg  8943  sbthlem8  9037  spthispth  28716
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