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Theorem f1ssres 6070
 Description: A function that is one-to-one is also one-to-one on some subset 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 6063 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fssres 6032 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
31, 2sylan 488 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
4 df-f1 5857 . . . . 5 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
54simprbi 480 . . . 4 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
6 funres11 5929 . . . 4 (Fun 𝐹 → Fun (𝐹𝐶))
75, 6syl 17 . . 3 (𝐹:𝐴1-1𝐵 → Fun (𝐹𝐶))
87adantr 481 . 2 ((𝐹:𝐴1-1𝐵𝐶𝐴) → Fun (𝐹𝐶))
9 df-f1 5857 . 2 ((𝐹𝐶):𝐶1-1𝐵 ↔ ((𝐹𝐶):𝐶𝐵 ∧ Fun (𝐹𝐶)))
103, 8, 9sylanbrc 697 1 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ⊆ wss 3559  ◡ccnv 5078   ↾ cres 5081  Fun wfun 5846  ⟶wf 5848  –1-1→wf1 5849 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857 This theorem is referenced by:  f1ores  6113  oacomf1olem  7596  pwfseqlem5  9437  hashimarn  13173  hashf1lem2  13186  conjsubgen  17625  sylow1lem2  17946  sylow2blem1  17967
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