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

Theorem fococnv2 5180
 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 5135 . . 3 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funcocnv2 5179 . . 3 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
31, 2syl 14 . 2 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))
4 forn 5137 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
54reseq2d 4640 . 2 (𝐹:𝐴onto𝐵 → ( I ↾ ran 𝐹) = ( I ↾ 𝐵))
63, 5eqtrd 2088 1 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259   I cid 4053  ◡ccnv 4372  ran crn 4374   ↾ cres 4375   ∘ ccom 4377  Fun wfun 4924  –onto→wfo 4928 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-fo 4936 This theorem is referenced by:  f1ococnv2  5181  foeqcnvco  5458
 Copyright terms: Public domain W3C validator