Theorem f1ococnv2 5487

Description: The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.)

Assertion

Ref	Expression
f1ococnv2	|- ( F : A -1-1-onto-> B -> ( F o. `' F ) = ( _I |` B ) )

Proof of Theorem f1ococnv2

Step	Hyp	Ref	Expression
1		f1ofo 5467	. 2 |- ( F : A -1-1-onto-> B -> F : A -onto-> B )
2		fococnv2 5486	. 2 |- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` B ) )
3	1, 2	syl 14	1 |- ( F : A -1-1-onto-> B -> ( F o. `' F ) = ( _I |` B ) )

Syntax hints:   -> wi 4   = wceq 1353   _I cid 4287   `'ccnv 4624   |` cres 4627   o. ccom 4629   -onto->wfo 5213   -1-1-onto->wf1o 5214

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222

This theorem is referenced by:  f1ococnv1  5489  f1ocnvfv2  5776  mapen  6843  hashfacen  10809