Theorem fococnv2 5773

Description: The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Assertion

Ref	Expression
fococnv2	|- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` B ) )

Proof of Theorem fococnv2

Step	Hyp	Ref	Expression
1		fofun 5728	. . 3 |- ( F : A -onto-> B -> Fun F )
2		funcocnv2 5772	. . 3 |- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) )
3	1, 2	syl 16	. 2 |- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` ran F ) )
4		forn 5730	. . 3 |- ( F : A -onto-> B -> ran F = B )
5	4	reseq2d 5217	. 2 |- ( F : A -onto-> B -> ( _I |` ran F ) = ( _I |` B ) )
6	3, 5	eqtrd 2495	1 |- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` B ) )

Syntax hints:   -> wi 4   = wceq 1370   _I cid 4738   `'ccnv 4946   ran crn 4948   |` cres 4949   o. ccom 4951   Fun wfun 5519   -onto->wfo 5523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400  df-opab 4458  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-fun 5527  df-fn 5528  df-f 5529  df-fo 5531

This theorem is referenced by:  f1ococnv2  5774  foeqcnvco  6106