Theorem fimacnvinrn 6348

Description: Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 21-Jan-2017.)

Assertion

Ref	Expression
fimacnvinrn	|- ( Fun F -> ( `' F " A ) = ( `' F " ( A i^i ran F ) ) )

Proof of Theorem fimacnvinrn

Step	Hyp	Ref	Expression
1		inpreima 6342	. 2 |- ( Fun F -> ( `' F " ( A i^i ran F ) ) = ( ( `' F " A ) i^i ( `' F " ran F ) ) )
2		funforn 6122	. . . . 5 |- ( Fun F <-> F : dom F -onto-> ran F )
3		fof 6115	. . . . 5 |- ( F : dom F -onto-> ran F -> F : dom F --> ran F )
4	2, 3	sylbi 207	. . . 4 |- ( Fun F -> F : dom F --> ran F )
5		fimacnv 6347	. . . 4 |- ( F : dom F --> ran F -> ( `' F " ran F ) = dom F )
6	4, 5	syl 17	. . 3 |- ( Fun F -> ( `' F " ran F ) = dom F )
7	6	ineq2d 3814	. 2 |- ( Fun F -> ( ( `' F " A ) i^i ( `' F " ran F ) ) = ( ( `' F " A ) i^i dom F ) )
8		cnvresima 5623	. . 3 |- ( `' ( F |` dom F ) " A ) = ( ( `' F " A ) i^i dom F )
9		resdm2 5624	. . . . . 6 |- ( F |` dom F ) = `' `' F
10		funrel 5905	. . . . . . 7 |- ( Fun F -> Rel F )
11		dfrel2 5583	. . . . . . 7 |- ( Rel F <-> `' `' F = F )
12	10, 11	sylib 208	. . . . . 6 |- ( Fun F -> `' `' F = F )
13	9, 12	syl5eq 2668	. . . . 5 |- ( Fun F -> ( F |` dom F ) = F )
14	13	cnveqd 5298	. . . 4 |- ( Fun F -> `' ( F |` dom F ) = `' F )
15	14	imaeq1d 5465	. . 3 |- ( Fun F -> ( `' ( F |` dom F ) " A ) = ( `' F " A ) )
16	8, 15	syl5eqr 2670	. 2 |- ( Fun F -> ( ( `' F " A ) i^i dom F ) = ( `' F " A ) )
17	1, 7, 16	3eqtrrd 2661	1 |- ( Fun F -> ( `' F " A ) = ( `' F " ( A i^i ran F ) ) )

Syntax hints:   -> wi 4   = wceq 1483   i^i cin 3573   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Rel wrel 5119   Fun wfun 5882   -->wf 5884   -onto->wfo 5886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896

This theorem is referenced by:  fimacnvinrn2  6349