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Theorem f1ssres 6344
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 6337 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fssres 6306 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
31, 2sylan 577 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
4 df-f1 6127 . . . . 5 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
54simprbi 492 . . . 4 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
6 funres11 6198 . . . 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 6127 . 2 ((𝐹𝐶):𝐶1-1𝐵 ↔ ((𝐹𝐶):𝐶𝐵 ∧ Fun (𝐹𝐶)))
103, 8, 9sylanbrc 580 1 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wss 3797  ccnv 5340  cres 5343  Fun wfun 6116  wf 6118  1-1wf1 6119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-br 4873  df-opab 4935  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127
This theorem is referenced by:  f1resf1  6345  f1ores  6391  oacomf1olem  7910  pwfseqlem5  9799  hashimarn  13515  hashf1lem2  13528  conjsubgen  18043  sylow1lem2  18364  sylow2blem1  18385  usgrres  26604
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