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Theorem f1f1orn 5388
 Description: A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
Assertion
Ref Expression
f1f1orn (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)

Proof of Theorem f1f1orn
StepHypRef Expression
1 f1fn 5340 . 2 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
2 df-f1 5138 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
32simprbi 273 . 2 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
4 f1orn 5387 . 2 (𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))
51, 3, 4sylanbrc 414 1 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4  ◡ccnv 4548  ran crn 4550  Fun wfun 5127   Fn wfn 5128  ⟶wf 5129  –1-1→wf1 5130  –1-1-onto→wf1o 5132 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3083  df-ss 3090  df-f 5137  df-f1 5138  df-fo 5139  df-f1o 5140 This theorem is referenced by:  f1ores  5392  f1cnv  5401  f1cocnv1  5407  f1ocnvfvrneq  5693  ssenen  6755  f1dmvrnfibi  6845  cc2lem  7121
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