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Theorem f1cocnv2 5504
Description: Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
f1cocnv2 (𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))

Proof of Theorem f1cocnv2
StepHypRef Expression
1 f1fun 5439 . 2 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
2 funcocnv2 5501 . 2 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
31, 2syl 14 1 (𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364   I cid 4303  ccnv 4640  ran crn 4642  cres 4643  ccom 4645  Fun wfun 5225  1-1wf1 5228
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236
This theorem is referenced by: (None)
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