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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 8230 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍))) | |
2 | 1 | imaeq2d 6080 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐺 “ (◡𝐺 “ (𝐹 supp 𝑍)))) |
3 | funforn 6828 | . . . 4 ⊢ (Fun 𝐺 ↔ 𝐺:dom 𝐺–onto→ran 𝐺) | |
4 | foimacnv 6866 | . . . 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 2795 | . 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺)) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍)) |
7 | 6 | ex 412 | 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 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ◡ccnv 5688 dom cdm 5689 ran crn 5690 “ cima 5692 ∘ ccom 5693 Fun wfun 6557 –onto→wfo 6561 (class class class)co 7431 supp csupp 8184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-supp 8185 |
This theorem is referenced by: gsumval3lem1 19938 gsumval3lem2 19939 |
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