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Theorem f1ssres 6349
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 6342 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fssres 6311 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
31, 2sylan 575 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
4 df-f1 6132 . . . . 5 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
54simprbi 492 . . . 4 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
6 funres11 6203 . . . 4 (Fun 𝐹 → Fun (𝐹𝐶))
75, 6syl 17 . . 3 (𝐹:𝐴1-1𝐵 → Fun (𝐹𝐶))
87adantr 474 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴) → Fun (𝐹𝐶))
9 df-f1 6132 . 2 ((𝐹𝐶):𝐶1-1𝐵 ↔ ((𝐹𝐶):𝐶𝐵 ∧ Fun (𝐹𝐶)))
103, 8, 9sylanbrc 578 1 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wss 3798  ccnv 5345  cres 5348  Fun wfun 6121  wf 6123  1-1wf1 6124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132
This theorem is referenced by:  f1resf1  6350  f1ores  6396  oacomf1olem  7916  pwfseqlem5  9807  hashimarn  13523  hashf1lem2  13536  conjsubgen  18051  sylow1lem2  18372  sylow2blem1  18393  usgrres  26612
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