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Theorem f1ssres 6575
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 6568 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fssres 6537 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
31, 2sylan 580 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
4 df-f1 6353 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
5 funres11 6424 . . . 4 (Fun 𝐹 → Fun (𝐹𝐶))
64, 5simplbiim 505 . . 3 (𝐹:𝐴1-1𝐵 → Fun (𝐹𝐶))
76adantr 481 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴) → Fun (𝐹𝐶))
8 df-f1 6353 . 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 396  wss 3933  ccnv 5547  cres 5550  Fun wfun 6342  wf 6344  1-1wf1 6345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353
This theorem is referenced by:  f1resf1  6576  f1ores  6622  oacomf1olem  8179  pwfseqlem5  10073  hashimarn  13789  hashf1lem2  13802  conjsubgen  18329  sylow1lem2  18653  sylow2blem1  18674  usgrres  27017
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