Theorem fococnv2 5661

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 5616	. . 3 |- ( F : A -onto-> B -> Fun F )
2		funcocnv2 5660	. . 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 5618	. . 3 |- ( F : A -onto-> B -> ran F = B )
5	4	reseq2d 5105	. 2 |- ( F : A -onto-> B -> ( _I |` ran F ) = ( _I |` B ) )
6	3, 5	eqtrd 2470	1 |- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` B ) )

Syntax hints:   -> wi 4   = wceq 1369   _I cid 4626   `'ccnv 4834   ran crn 4836   |` cres 4837   o. ccom 4839   Fun wfun 5407   -onto->wfo 5411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419

This theorem is referenced by:  f1ococnv2  5662  foeqcnvco  5993