Theorem f1ococnv2 5598

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 5578	. 2 |- ( F : A -1-1-onto-> B -> F : A -onto-> B )
2		fococnv2 5597	. 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 1395   _I cid 4378   `'ccnv 4717   |` cres 4720   o. ccom 4722   -onto->wfo 5315   -1-1-onto->wf1o 5316

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324

This theorem is referenced by:  f1ococnv1  5600  f1ocnvfv2  5901  mapen  7003  hashfacen  11053