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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 7948 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍))) | |
2 | 1 | imaeq2d 5929 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐺 “ (◡𝐺 “ (𝐹 supp 𝑍)))) |
3 | funforn 6640 | . . . 4 ⊢ (Fun 𝐺 ↔ 𝐺:dom 𝐺–onto→ran 𝐺) | |
4 | foimacnv 6678 | . . . 4 ⊢ ((𝐺:dom 𝐺–onto→ran 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ (◡𝐺 “ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍)) | |
5 | 3, 4 | sylanb 584 | . . 3 ⊢ ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ (◡𝐺 “ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍)) |
6 | 2, 5 | sylan9eq 2798 | . 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍)) |
7 | 6 | ex 416 | 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 399 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 ◡ccnv 5550 dom cdm 5551 ran crn 5552 “ cima 5554 ∘ ccom 5555 Fun wfun 6374 –onto→wfo 6378 (class class class)co 7213 supp csupp 7903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fo 6386 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-supp 7904 |
This theorem is referenced by: gsumval3lem1 19290 gsumval3lem2 19291 |
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