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Mirrors > Home > MPE Home > Th. List > fczsupp0 | Structured version Visualization version GIF version |
Description: The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.) |
Ref | Expression |
---|---|
fczsupp0 | ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2727 | . . 3 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) = (𝐵 × {𝑍})) | |
2 | fnconstg 6792 | . . . . 5 ⊢ (𝑍 ∈ V → (𝐵 × {𝑍}) Fn 𝐵) | |
3 | 2 | adantl 480 | . . . 4 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) Fn 𝐵) |
4 | snnzg 4783 | . . . . 5 ⊢ (𝑍 ∈ V → {𝑍} ≠ ∅) | |
5 | simpl 481 | . . . . 5 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) ∈ V) | |
6 | xpexcnv 7935 | . . . . 5 ⊢ (({𝑍} ≠ ∅ ∧ (𝐵 × {𝑍}) ∈ V) → 𝐵 ∈ V) | |
7 | 4, 5, 6 | syl2an2 684 | . . . 4 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → 𝐵 ∈ V) |
8 | simpr 483 | . . . 4 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V) | |
9 | fnsuppeq0 8208 | . . . 4 ⊢ (((𝐵 × {𝑍}) Fn 𝐵 ∧ 𝐵 ∈ V ∧ 𝑍 ∈ V) → (((𝐵 × {𝑍}) supp 𝑍) = ∅ ↔ (𝐵 × {𝑍}) = (𝐵 × {𝑍}))) | |
10 | 3, 7, 8, 9 | syl3anc 1368 | . . 3 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (((𝐵 × {𝑍}) supp 𝑍) = ∅ ↔ (𝐵 × {𝑍}) = (𝐵 × {𝑍}))) |
11 | 1, 10 | mpbird 256 | . 2 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → ((𝐵 × {𝑍}) supp 𝑍) = ∅) |
12 | supp0prc 8179 | . 2 ⊢ (¬ ((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → ((𝐵 × {𝑍}) supp 𝑍) = ∅) | |
13 | 11, 12 | pm2.61i 182 | 1 ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 ∅c0 4325 {csn 4633 × cxp 5682 Fn wfn 6551 (class class class)co 7426 supp csupp 8176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pr 5435 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-ov 7429 df-oprab 7430 df-mpo 7431 df-supp 8177 |
This theorem is referenced by: fczfsuppd 9431 cantnf 9738 mhp0cl 22142 cantnfresb 43008 |
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