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Mirrors > Home > MPE Home > Th. List > Mathboxes > supppreima | Structured version Visualization version GIF version |
Description: Express the support of a function as the preimage of its range except zero. (Contributed by Thierry Arnoux, 24-Jun-2024.) |
Ref | Expression |
---|---|
supppreima | ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (ran 𝐹 ∖ {𝑍}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvimarndm 5939 | . . . 4 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹 | |
2 | 1 | a1i 11 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (◡𝐹 “ ran 𝐹) = dom 𝐹) |
3 | 2 | difeq1d 4026 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((◡𝐹 “ ran 𝐹) ∖ (◡𝐹 “ {𝑍})) = (dom 𝐹 ∖ (◡𝐹 “ {𝑍}))) |
4 | difpreima 6874 | . . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∖ {𝑍})) = ((◡𝐹 “ ran 𝐹) ∖ (◡𝐹 “ {𝑍}))) | |
5 | 4 | 3ad2ant1 1135 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (◡𝐹 “ (ran 𝐹 ∖ {𝑍})) = ((◡𝐹 “ ran 𝐹) ∖ (◡𝐹 “ {𝑍}))) |
6 | suppssdm 7908 | . . . 4 ⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹 | |
7 | dfss4 4163 | . . . 4 ⊢ ((𝐹 supp 𝑍) ⊆ dom 𝐹 ↔ (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍)) | |
8 | 6, 7 | mpbi 233 | . . 3 ⊢ (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 supp 𝑍) |
9 | suppiniseg 30712 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍})) | |
10 | 9 | difeq2d 4027 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (dom 𝐹 ∖ (◡𝐹 “ {𝑍}))) |
11 | 8, 10 | eqtr3id 2788 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (dom 𝐹 ∖ (◡𝐹 “ {𝑍}))) |
12 | 3, 5, 11 | 3eqtr4rd 2785 | 1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (ran 𝐹 ∖ {𝑍}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∖ cdif 3854 ⊆ wss 3857 {csn 4531 ◡ccnv 5539 dom cdm 5540 ran crn 5541 “ cima 5543 Fun wfun 6363 (class class class)co 7202 supp csupp 7892 |
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 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512 |
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 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-supp 7893 |
This theorem is referenced by: gsumhashmul 31007 |
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