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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 3920 | . . . 4 ⊢ (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍))) | |
2 | funfn 6531 | . . . . . . . . . . 11 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
3 | 2 | biimpi 215 | . . . . . . . . . 10 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹) |
4 | elsuppfng 8100 | . . . . . . . . . 10 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 𝑍))) | |
5 | 3, 4 | syl3an1 1163 | . . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ≠ 𝑍))) |
6 | 5 | baibd 540 | . . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) ≠ 𝑍)) |
7 | 6 | notbid 317 | . . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹‘𝑥) ≠ 𝑍)) |
8 | nne 2947 | . . . . . . 7 ⊢ (¬ (𝐹‘𝑥) ≠ 𝑍 ↔ (𝐹‘𝑥) = 𝑍) | |
9 | 7, 8 | bitrdi 286 | . . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) = 𝑍)) |
10 | fvex 6855 | . . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V | |
11 | 10 | elsn 4601 | . . . . . 6 ⊢ ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍) |
12 | 9, 11 | bitr4di 288 | . . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) ∈ {𝑍})) |
13 | 12 | pm5.32da 579 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝑥 ∈ dom 𝐹 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ {𝑍}))) |
14 | 1, 13 | bitrid 282 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ {𝑍}))) |
15 | 3 | 3ad2ant1 1133 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐹 Fn dom 𝐹) |
16 | elpreima 7008 | . . . 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 281 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑥 ∈ (dom 𝐹 ∖ (𝐹 supp 𝑍)) ↔ 𝑥 ∈ (◡𝐹 “ {𝑍}))) |
19 | 18 | eqrdv 2734 | 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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3907 {csn 4586 ◡ccnv 5632 dom cdm 5633 “ cima 5636 Fun wfun 6490 Fn wfn 6491 ‘cfv 6496 (class class class)co 7356 supp csupp 8091 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7671 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-fv 6504 df-ov 7359 df-oprab 7360 df-mpo 7361 df-supp 8092 |
This theorem is referenced by: fressupp 31548 supppreima 31551 |
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