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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 7066 | . 2 β’ (Fun πΉ β (β‘πΉ β (π΄ β© ran πΉ)) = ((β‘πΉ β π΄) β© (β‘πΉ β ran πΉ))) | |
2 | funforn 6813 | . . . . 5 β’ (Fun πΉ β πΉ:dom πΉβontoβran πΉ) | |
3 | fof 6806 | . . . . 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 6740 | . . . 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 4213 | . 2 β’ (Fun πΉ β ((β‘πΉ β π΄) β© (β‘πΉ β ran πΉ)) = ((β‘πΉ β π΄) β© dom πΉ)) |
8 | cnvresima 6230 | . . 3 β’ (β‘(πΉ βΎ dom πΉ) β π΄) = ((β‘πΉ β π΄) β© dom πΉ) | |
9 | resdm2 6231 | . . . . . 6 β’ (πΉ βΎ dom πΉ) = β‘β‘πΉ | |
10 | funrel 6566 | . . . . . . 7 β’ (Fun πΉ β Rel πΉ) | |
11 | dfrel2 6189 | . . . . . . 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 2785 | . . . . 5 β’ (Fun πΉ β (πΉ βΎ dom πΉ) = πΉ) |
14 | 13 | cnveqd 5876 | . . . 4 β’ (Fun πΉ β β‘(πΉ βΎ dom πΉ) = β‘πΉ) |
15 | 14 | imaeq1d 6059 | . . 3 β’ (Fun πΉ β (β‘(πΉ βΎ dom πΉ) β π΄) = (β‘πΉ β π΄)) |
16 | 8, 15 | eqtr3id 2787 | . 2 β’ (Fun πΉ β ((β‘πΉ β π΄) β© dom πΉ) = (β‘πΉ β π΄)) |
17 | 1, 7, 16 | 3eqtrrd 2778 | 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 1542 β© cin 3948 β‘ccnv 5676 dom cdm 5677 ran crn 5678 βΎ cres 5679 β cima 5680 Rel wrel 5682 Fun wfun 6538 βΆwf 6540 βontoβwfo 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-fun 6546 df-fn 6547 df-f 6548 df-fo 6550 |
This theorem is referenced by: fimacnvinrn2 7075 preiman0 31931 |
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