MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1cocnv1 Structured version   Visualization version   GIF version

Theorem f1cocnv1 6819
Description: Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
f1cocnv1 (𝐹:𝐴–1-1→𝐡 β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ 𝐴))

Proof of Theorem f1cocnv1
StepHypRef Expression
1 f1f1orn 6800 . 2 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
2 f1ococnv1 6818 . 2 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ 𝐴))
31, 2syl 17 1 (𝐹:𝐴–1-1→𝐡 β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   I cid 5535  β—‘ccnv 5637  ran crn 5639   β†Ύ cres 5640   ∘ ccom 5642  β€“1-1β†’wf1 6498  β€“1-1-ontoβ†’wf1o 6500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508
This theorem is referenced by:  f1eqcocnv  7252  f1eqcocnvOLD  7253  domss2  9087  diophrw  41111
  Copyright terms: Public domain W3C validator