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Theorem funres11 4914
 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 4578 . 2 (𝐹A) ⊆ 𝐹
2 cnvss 4451 . 2 ((𝐹A) ⊆ 𝐹(𝐹A) ⊆ 𝐹)
3 funss 4863 . 2 ((𝐹A) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹A)))
41, 2, 3mp2b 8 1 (Fun 𝐹 → Fun (𝐹A))
 Colors of variables: wff set class Syntax hints:   → wi 4   ⊆ wss 2911  ◡ccnv 4287   ↾ cres 4290  Fun wfun 4839 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-res 4300  df-fun 4847 This theorem is referenced by:  f1ssres  5042  resdif  5091  ssdomg  6194
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