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Theorem funres11 6619
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 6000 . 2 (𝐹𝐴) ⊆ 𝐹
2 cnvss 5866 . 2 ((𝐹𝐴) ⊆ 𝐹(𝐹𝐴) ⊆ 𝐹)
3 funss 6561 . 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 3943  ccnv 5668  cres 5671  Fun wfun 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-br 5142  df-opab 5204  df-rel 5676  df-cnv 5677  df-co 5678  df-res 5681  df-fun 6539
This theorem is referenced by:  f1ssres  6789  resdif  6848  f1ssf1  6859  ssdomg  8998  sbthlem8  9092  spthispth  29492
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