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Theorem f1ococnv1 5505
Description: The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.)
Assertion
Ref Expression
f1ococnv1 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))

Proof of Theorem f1ococnv1
StepHypRef Expression
1 f1orel 5479 . . . 4 (𝐹:𝐴1-1-onto𝐵 → Rel 𝐹)
2 dfrel2 5094 . . . 4 (Rel 𝐹𝐹 = 𝐹)
31, 2sylib 122 . . 3 (𝐹:𝐴1-1-onto𝐵𝐹 = 𝐹)
43coeq2d 4804 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = (𝐹𝐹))
5 f1ocnv 5489 . . 3 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
6 f1ococnv2 5503 . . 3 (𝐹:𝐵1-1-onto𝐴 → (𝐹𝐹) = ( I ↾ 𝐴))
75, 6syl 14 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
84, 7eqtr3d 2224 1 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364   I cid 4303  ccnv 4640  cres 4643  ccom 4645  Rel wrel 4646  1-1-ontowf1o 5230
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  df-fo 5237  df-f1o 5238
This theorem is referenced by:  f1cocnv1  5506  f1ocnvfv1  5794  fcof1o  5806  mapen  6864  hashfacen  10835
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