Theorem f1ococnv2 5610

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 5590	. 2 |- ( F : A -1-1-onto-> B -> F : A -onto-> B )
2		fococnv2 5609	. 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 1397   _I cid 4385   `'ccnv 4724   |` cres 4727   o. ccom 4729   -onto->wfo 5324   -1-1-onto->wf1o 5325

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333

This theorem is referenced by:  f1ococnv1  5612  f1ocnvfv2  5918  mapen  7031  hashfacen  11099