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Mirrors > Home > MPE Home > Th. List > imacosupp | Structured version Visualization version GIF version |
Description: The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.) |
Ref | Expression |
---|---|
imacosupp | β’ ((πΉ β π β§ πΊ β π) β ((Fun πΊ β§ (πΉ supp π) β ran πΊ) β (πΊ β ((πΉ β πΊ) supp π)) = (πΉ supp π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppco 8141 | . . . 4 β’ ((πΉ β π β§ πΊ β π) β ((πΉ β πΊ) supp π) = (β‘πΊ β (πΉ supp π))) | |
2 | 1 | imaeq2d 6017 | . . 3 β’ ((πΉ β π β§ πΊ β π) β (πΊ β ((πΉ β πΊ) supp π)) = (πΊ β (β‘πΊ β (πΉ supp π)))) |
3 | funforn 6767 | . . . 4 β’ (Fun πΊ β πΊ:dom πΊβontoβran πΊ) | |
4 | foimacnv 6805 | . . . 4 β’ ((πΊ:dom πΊβontoβran πΊ β§ (πΉ supp π) β ran πΊ) β (πΊ β (β‘πΊ β (πΉ supp π))) = (πΉ supp π)) | |
5 | 3, 4 | sylanb 582 | . . 3 β’ ((Fun πΊ β§ (πΉ supp π) β ran πΊ) β (πΊ β (β‘πΊ β (πΉ supp π))) = (πΉ supp π)) |
6 | 2, 5 | sylan9eq 2793 | . 2 β’ (((πΉ β π β§ πΊ β π) β§ (Fun πΊ β§ (πΉ supp π) β ran πΊ)) β (πΊ β ((πΉ β πΊ) supp π)) = (πΉ supp π)) |
7 | 6 | ex 414 | 1 β’ ((πΉ β π β§ πΊ β π) β ((Fun πΊ β§ (πΉ supp π) β ran πΊ) β (πΊ β ((πΉ β πΊ) supp π)) = (πΉ supp π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3914 β‘ccnv 5636 dom cdm 5637 ran crn 5638 β cima 5640 β ccom 5641 Fun wfun 6494 βontoβwfo 6498 (class class class)co 7361 supp csupp 8096 |
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-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fo 6506 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-supp 8097 |
This theorem is referenced by: gsumval3lem1 19690 gsumval3lem2 19691 |
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