Theorem f1ococnv1 3715

Description: The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain.

Assertion

Ref	Expression
f1ococnv1	|- (F:A-1-1-onto->B -> (`'F o. F) = (I |` A))

Proof of Theorem f1ococnv1

Step	Hyp	Ref	Expression
1		f1orel 3698	. . . 4 |- (F:A-1-1-onto->B -> Rel F)
2		dfrel2 3491	. . . 4 |- (Rel F <-> `'`'F = F)
3	1, 2	sylib 198	. . 3 |- (F:A-1-1-onto->B -> `'`'F = F)
4	3	coeq2d 3292	. 2 |- (F:A-1-1-onto->B -> (`'F o. `'`'F) = (`'F o. F))
5		f1ocnv 3707	. . 3 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
6		f1ococnv2 3714	. . 3 |- (`'F:B-1-1-onto->A -> (`'F o. `'`'F) = (I |` A))
7	5, 6	syl 10	. 2 |- (F:A-1-1-onto->B -> (`'F o. `'`'F) = (I |` A))
8	4, 7	eqtr3d 1512	1 |- (F:A-1-1-onto->B -> (`'F o. F) = (I |` A))

Syntax hints:   -> wi 3   = wceq 958  Icid 2837  `'ccnv 3175   |` cres 3178   o. ccom 3180  Rel wrel 3181  -1-1-onto->wf1o 3187

This theorem is referenced by:  f1ocnvfv1 3884  mapenlem1 4495  adjbdlnb 10012  symggrpi 10401  hmeogrp 10524

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203

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