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Mirrors > Home > MPE Home > Th. List > fimacnvinrn | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fimacnvinrn | β’ (Fun πΉ β (β‘πΉ β π΄) = (β‘πΉ β (π΄ β© ran πΉ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inpreima 7064 | . 2 β’ (Fun πΉ β (β‘πΉ β (π΄ β© ran πΉ)) = ((β‘πΉ β π΄) β© (β‘πΉ β ran πΉ))) | |
2 | funforn 6811 | . . . . 5 β’ (Fun πΉ β πΉ:dom πΉβontoβran πΉ) | |
3 | fof 6804 | . . . . 5 β’ (πΉ:dom πΉβontoβran πΉ β πΉ:dom πΉβΆran πΉ) | |
4 | 2, 3 | sylbi 216 | . . . 4 β’ (Fun πΉ β πΉ:dom πΉβΆran πΉ) |
5 | fimacnv 6738 | . . . 4 β’ (πΉ:dom πΉβΆran πΉ β (β‘πΉ β ran πΉ) = dom πΉ) | |
6 | 4, 5 | syl 17 | . . 3 β’ (Fun πΉ β (β‘πΉ β ran πΉ) = dom πΉ) |
7 | 6 | ineq2d 4211 | . 2 β’ (Fun πΉ β ((β‘πΉ β π΄) β© (β‘πΉ β ran πΉ)) = ((β‘πΉ β π΄) β© dom πΉ)) |
8 | cnvresima 6228 | . . 3 β’ (β‘(πΉ βΎ dom πΉ) β π΄) = ((β‘πΉ β π΄) β© dom πΉ) | |
9 | resdm2 6229 | . . . . . 6 β’ (πΉ βΎ dom πΉ) = β‘β‘πΉ | |
10 | funrel 6564 | . . . . . . 7 β’ (Fun πΉ β Rel πΉ) | |
11 | dfrel2 6187 | . . . . . . 7 β’ (Rel πΉ β β‘β‘πΉ = πΉ) | |
12 | 10, 11 | sylib 217 | . . . . . 6 β’ (Fun πΉ β β‘β‘πΉ = πΉ) |
13 | 9, 12 | eqtrid 2782 | . . . . 5 β’ (Fun πΉ β (πΉ βΎ dom πΉ) = πΉ) |
14 | 13 | cnveqd 5874 | . . . 4 β’ (Fun πΉ β β‘(πΉ βΎ dom πΉ) = β‘πΉ) |
15 | 14 | imaeq1d 6057 | . . 3 β’ (Fun πΉ β (β‘(πΉ βΎ dom πΉ) β π΄) = (β‘πΉ β π΄)) |
16 | 8, 15 | eqtr3id 2784 | . 2 β’ (Fun πΉ β ((β‘πΉ β π΄) β© dom πΉ) = (β‘πΉ β π΄)) |
17 | 1, 7, 16 | 3eqtrrd 2775 | 1 β’ (Fun πΉ β (β‘πΉ β π΄) = (β‘πΉ β (π΄ β© ran πΉ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β© cin 3946 β‘ccnv 5674 dom cdm 5675 ran crn 5676 βΎ cres 5677 β cima 5678 Rel wrel 5680 Fun wfun 6536 βΆwf 6538 βontoβwfo 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-fun 6544 df-fn 6545 df-f 6546 df-fo 6548 |
This theorem is referenced by: fimacnvinrn2 7073 preiman0 32198 |
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