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Theorem fococnv2 5361
Description: The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
Assertion
Ref Expression
fococnv2 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))

Proof of Theorem fococnv2
StepHypRef Expression
1 fofun 5316 . . 3 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funcocnv2 5360 . . 3 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
31, 2syl 14 . 2 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))
4 forn 5318 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
54reseq2d 4789 . 2 (𝐹:𝐴onto𝐵 → ( I ↾ ran 𝐹) = ( I ↾ 𝐵))
63, 5eqtrd 2150 1 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316   I cid 4180  ccnv 4508  ran crn 4510  cres 4511  ccom 4513  Fun wfun 5087  ontowfo 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099
This theorem is referenced by:  f1ococnv2  5362  foeqcnvco  5659
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