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Theorem f1ssres 6812
Description: A function that is one-to-one is also one-to-one on any subclass of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)
Assertion
Ref Expression
f1ssres ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1𝐵)

Proof of Theorem f1ssres
StepHypRef Expression
1 f1f 6805 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fssres 6775 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
31, 2sylan 580 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
4 df-f1 6568 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
5 funres11 6645 . . . 4 (Fun 𝐹 → Fun (𝐹𝐶))
64, 5simplbiim 504 . . 3 (𝐹:𝐴1-1𝐵 → Fun (𝐹𝐶))
76adantr 480 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴) → Fun (𝐹𝐶))
8 df-f1 6568 . 2 ((𝐹𝐶):𝐶1-1𝐵 ↔ ((𝐹𝐶):𝐶𝐵 ∧ Fun (𝐹𝐶)))
93, 7, 8sylanbrc 583 1 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wss 3963  ccnv 5688  cres 5691  Fun wfun 6557  wf 6559  1-1wf1 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568
This theorem is referenced by:  f1resf1  6813  f1ores  6863  oacomf1olem  8601  domssl  9037  undom  9098  pwfseqlem5  10701  hashimarn  14476  hashf1lem2  14492  conjsubgen  19282  sylow1lem2  19632  sylow2blem1  19653  usgrres  29340  lmimdim  33631
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