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Theorem f1ococnv1 5183
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 5157 . . . 4 (𝐹:𝐴1-1-onto𝐵 → Rel 𝐹)
2 dfrel2 4799 . . . 4 (Rel 𝐹𝐹 = 𝐹)
31, 2sylib 131 . . 3 (𝐹:𝐴1-1-onto𝐵𝐹 = 𝐹)
43coeq2d 4526 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = (𝐹𝐹))
5 f1ocnv 5167 . . 3 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
6 f1ococnv2 5181 . . 3 (𝐹:𝐵1-1-onto𝐴 → (𝐹𝐹) = ( I ↾ 𝐴))
75, 6syl 14 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
84, 7eqtr3d 2090 1 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259   I cid 4053  ccnv 4372  cres 4375  ccom 4377  Rel wrel 4378  1-1-ontowf1o 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937
This theorem is referenced by:  f1cocnv1  5184  f1ocnvfv1  5445  fcof1o  5457
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