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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 | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppco 8173 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍))) | |
2 | 1 | imaeq2d 6049 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐺 “ (◡𝐺 “ (𝐹 supp 𝑍)))) |
3 | funforn 6799 | . . . 4 ⊢ (Fun 𝐺 ↔ 𝐺:dom 𝐺–onto→ran 𝐺) | |
4 | foimacnv 6837 | . . . 4 ⊢ ((𝐺:dom 𝐺–onto→ran 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ (◡𝐺 “ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍)) | |
5 | 3, 4 | sylanb 581 | . . 3 ⊢ ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ (◡𝐺 “ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍)) |
6 | 2, 5 | sylan9eq 2791 | . 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍)) |
7 | 6 | ex 413 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3944 ◡ccnv 5668 dom cdm 5669 ran crn 5670 “ cima 5672 ∘ ccom 5673 Fun wfun 6526 –onto→wfo 6530 (class class class)co 7393 supp csupp 8128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fo 6538 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-supp 8129 |
This theorem is referenced by: gsumval3lem1 19732 gsumval3lem2 19733 |
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