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Theorem f1f1orn 5474
Description: A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
Assertion
Ref Expression
f1f1orn (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)

Proof of Theorem f1f1orn
StepHypRef Expression
1 f1fn 5425 . 2 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹 Fn 𝐴)
2 df-f1 5223 . . 3 (𝐹:𝐴–1-1→𝐡 ↔ (𝐹:𝐴⟢𝐡 ∧ Fun ◑𝐹))
32simprbi 275 . 2 (𝐹:𝐴–1-1→𝐡 β†’ Fun ◑𝐹)
4 f1orn 5473 . 2 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◑𝐹))
51, 3, 4sylanbrc 417 1 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4  β—‘ccnv 4627  ran crn 4629  Fun wfun 5212   Fn wfn 5213  βŸΆwf 5214  β€“1-1β†’wf1 5215  β€“1-1-ontoβ†’wf1o 5217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225
This theorem is referenced by:  f1ores  5478  f1cnv  5487  f1cocnv1  5493  f1ocnvfvrneq  5786  ssenen  6854  f1dmvrnfibi  6946  cc2lem  7268  xpsff1o2  12776
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