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Theorem f1ssres 5042
Description: A function that is one-to-one is also one-to-one on some aubset of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)
Assertion
Ref Expression
f1ssres ((𝐹:A1-1B 𝐶A) → (𝐹𝐶):𝐶1-1B)

Proof of Theorem f1ssres
StepHypRef Expression
1 f1f 5035 . . 3 (𝐹:A1-1B𝐹:AB)
2 fssres 5009 . . 3 ((𝐹:AB 𝐶A) → (𝐹𝐶):𝐶B)
31, 2sylan 267 . 2 ((𝐹:A1-1B 𝐶A) → (𝐹𝐶):𝐶B)
4 df-f1 4850 . . . . 5 (𝐹:A1-1B ↔ (𝐹:AB Fun 𝐹))
54simprbi 260 . . . 4 (𝐹:A1-1B → Fun 𝐹)
6 funres11 4914 . . . 4 (Fun 𝐹 → Fun (𝐹𝐶))
75, 6syl 14 . . 3 (𝐹:A1-1B → Fun (𝐹𝐶))
87adantr 261 . 2 ((𝐹:A1-1B 𝐶A) → Fun (𝐹𝐶))
9 df-f1 4850 . 2 ((𝐹𝐶):𝐶1-1B ↔ ((𝐹𝐶):𝐶B Fun (𝐹𝐶)))
103, 8, 9sylanbrc 394 1 ((𝐹:A1-1B 𝐶A) → (𝐹𝐶):𝐶1-1B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wss 2911  ccnv 4287  cres 4290  Fun wfun 4839  wf 4841  1-1wf1 4842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850
This theorem is referenced by:  f1ores  5084
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