Theorem f1ococnv2 3708

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

Assertion

Ref	Expression
f1ococnv2	|- (F:A-1-1-onto->B -> (F o. `'F) = (I |` B))

Proof of Theorem f1ococnv2

Step	Hyp	Ref	Expression
1		f1of 3689	. . . 4 |- (F:A-1-1-onto->B -> F:A-->B)
2		ffun 3629	. . . 4 |- (F:A-->B -> Fun F)
3		df-fun 3192	. . . . 5 |- (Fun F <-> (Rel F /\ (F o. `'F) (_ I))
4	3	pm3.27bi 326	. . . 4 |- (Fun F -> (F o. `'F) (_ I)
5	1, 2, 4	3syl 20	. . 3 |- (F:A-1-1-onto->B -> (F o. `'F) (_ I)
6		iss 3397	. . 3 |- ((F o. `'F) (_ I <-> (F o. `'F) = (I |` dom ( F o. `'F)))
7	5, 6	sylib 198	. 2 |- (F:A-1-1-onto->B -> (F o. `'F) = (I |` dom ( F o. `'F)))
8		fdm 3631	. . . . . . 7 |- (F:A-->B -> dom F = A)
9	1, 8	syl 10	. . . . . 6 |- (F:A-1-1-onto->B -> dom F = A)
10		f1ocnv 3701	. . . . . . 7 |- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
11		f1ofo 3695	. . . . . . 7 |- (`'F:B-1-1-onto->A -> `'F:B-onto->A)
12		forn 3674	. . . . . . 7 |- (`'F:B-onto->A -> ran `' F = A)
13	10, 11, 12	3syl 20	. . . . . 6 |- (F:A-1-1-onto->B -> ran `' F = A)
14	9, 13	eqtr4d 1510	. . . . 5 |- (F:A-1-1-onto->B -> dom F = ran `' F)
15		dmcoeq 3366	. . . . 5 |- (dom F = ran `' F -> dom ( F o. `'F) = dom `' F)
16	14, 15	syl 10	. . . 4 |- (F:A-1-1-onto->B -> dom ( F o. `'F) = dom `' F)
17		f1of 3689	. . . . 5 |- (`'F:B-1-1-onto->A -> `'F:B-->A)
18		fdm 3631	. . . . 5 |- (`'F:B-->A -> dom `' F = B)
19	10, 17, 18	3syl 20	. . . 4 |- (F:A-1-1-onto->B -> dom `' F = B)
20	16, 19	eqtrd 1507	. . 3 |- (F:A-1-1-onto->B -> dom ( F o. `'F) = B)
21		reseq2 3369	. . 3 |- (dom ( F o. `'F) = B -> (I |` dom ( F o. `'F)) = (I |` B))
22	20, 21	syl 10	. 2 |- (F:A-1-1-onto->B -> (I |` dom ( F o. `'F)) = (I |` B))
23	7, 22	eqtrd 1507	1 |- (F:A-1-1-onto->B -> (F o. `'F) = (I |` B))

Syntax hints:   -> wi 3   = wceq 956   (_ wss 2047  Icid 2831  `'ccnv 3169  dom cdm 3170  ran crn 3171   |` cres 3172   o. ccom 3174  Rel wrel 3175  Fun wfun 3176  -->wf 3178  -onto->wfo 3180  -1-1-onto->wf1o 3181

This theorem is referenced by:  f1ococnv1 3709  f1ocnvfv2 3879  mapenlem2 4490

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197

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