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Mirrors > Home > MPE Home > Th. List > Mathboxes > suppiniseg | Structured version Visualization version GIF version |
Description: Relation between the support (𝐹 supp 𝑍) and the initial segment (◡𝐹 “ {𝑍}). (Contributed by Thierry Arnoux, 25-Jun-2024.) |
Ref | Expression |
---|---|
suppiniseg | ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3867 | . . . 4 ⊢ (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍))) | |
2 | funfn 6399 | . . . . . . . . . . 11 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
3 | 2 | biimpi 219 | . . . . . . . . . 10 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹) |
4 | elsuppfng 7901 | . . . . . . . . . 10 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 𝑍))) | |
5 | 3, 4 | syl3an1 1165 | . . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 𝑍))) |
6 | 5 | baibd 543 | . . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) ≠ 𝑍)) |
7 | 6 | notbid 321 | . . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹‘𝑥) ≠ 𝑍)) |
8 | nne 2939 | . . . . . . 7 ⊢ (¬ (𝐹‘𝑥) ≠ 𝑍 ↔ (𝐹‘𝑥) = 𝑍) | |
9 | 7, 8 | bitrdi 290 | . . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) = 𝑍)) |
10 | fvex 6719 | . . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V | |
11 | 10 | elsn 4546 | . . . . . 6 ⊢ ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍) |
12 | 9, 11 | bitr4di 292 | . . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) ∈ {𝑍})) |
13 | 12 | pm5.32da 582 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ {𝑍}))) |
14 | 1, 13 | syl5bb 286 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ {𝑍}))) |
15 | 3 | 3ad2ant1 1135 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐹 Fn dom 𝐹) |
16 | elpreima 6867 | . . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ {𝑍}) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ {𝑍}))) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (◡𝐹 “ {𝑍}) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ {𝑍}))) |
18 | 14, 17 | bitr4d 285 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ 𝑥 ∈ (◡𝐹 “ {𝑍}))) |
19 | 18 | eqrdv 2732 | 1 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∖ cdif 3854 {csn 4531 ◡ccnv 5539 dom cdm 5540 “ cima 5543 Fun wfun 6363 Fn wfn 6364 ‘cfv 6369 (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: fressupp 30714 supppreima 30717 |
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