MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1cocnv2 Structured version   Visualization version   GIF version

Theorem f1cocnv2 6305
Description: Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
f1cocnv2 (𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))

Proof of Theorem f1cocnv2
StepHypRef Expression
1 f1fun 6243 . 2 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
2 funcocnv2 6302 . 2 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
31, 2syl 17 1 (𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630   I cid 5156  ccnv 5248  ran crn 5250  cres 5251  ccom 5253  Fun wfun 6025  1-1wf1 6028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator