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Theorem f1cocnv1 5503
Description: Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
f1cocnv1 (𝐹:𝐴–1-1→𝐡 β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ 𝐴))

Proof of Theorem f1cocnv1
StepHypRef Expression
1 f1f1orn 5484 . 2 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
2 f1ococnv1 5502 . 2 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ 𝐴))
31, 2syl 14 1 (𝐹:𝐴–1-1→𝐡 β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ 𝐴))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   = wceq 1363   I cid 4300  β—‘ccnv 4637  ran crn 4639   β†Ύ cres 4640   ∘ ccom 4642  β€“1-1β†’wf1 5225  β€“1-1-ontoβ†’wf1o 5227
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235
This theorem is referenced by:  f1eqcocnv  5805
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