Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fococnv2 Structured version   Visualization version   GIF version

Theorem fococnv2 6402
 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 6353 . . 3 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funcocnv2 6401 . . 3 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
31, 2syl 17 . 2 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))
4 forn 6355 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
54reseq2d 5628 . 2 (𝐹:𝐴onto𝐵 → ( I ↾ ran 𝐹) = ( I ↾ 𝐵))
63, 5eqtrd 2860 1 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   I cid 5248  ◡ccnv 5340  ran crn 5342   ↾ cres 5343   ∘ ccom 5345  Fun wfun 6116  –onto→wfo 6120 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-br 4873  df-opab 4935  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-fun 6124  df-fn 6125  df-f 6126  df-fo 6128 This theorem is referenced by:  f1ococnv2  6403  foeqcnvco  6809
 Copyright terms: Public domain W3C validator