ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fococnv2 GIF version

Theorem fococnv2 5526
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 5477 . . 3 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funcocnv2 5525 . . 3 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
31, 2syl 14 . 2 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))
4 forn 5479 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
54reseq2d 4942 . 2 (𝐹:𝐴onto𝐵 → ( I ↾ ran 𝐹) = ( I ↾ 𝐵))
63, 5eqtrd 2226 1 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364   I cid 4319  ccnv 4658  ran crn 4660  cres 4661  ccom 4663  Fun wfun 5248  ontowfo 5252
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 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-fun 5256  df-fn 5257  df-f 5258  df-fo 5260
This theorem is referenced by:  f1ococnv2  5527  foeqcnvco  5833
  Copyright terms: Public domain W3C validator