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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 2773 | . . 3 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) = (𝐵 × {𝑍})) | |
2 | fnconstg 6390 | . . . . 5 ⊢ (𝑍 ∈ V → (𝐵 × {𝑍}) Fn 𝐵) | |
3 | 2 | adantl 474 | . . . 4 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) Fn 𝐵) |
4 | snnzg 4578 | . . . . 5 ⊢ (𝑍 ∈ V → {𝑍} ≠ ∅) | |
5 | simpl 475 | . . . . 5 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) ∈ V) | |
6 | xpexcnv 7434 | . . . . 5 ⊢ (({𝑍} ≠ ∅ ∧ (𝐵 × {𝑍}) ∈ V) → 𝐵 ∈ V) | |
7 | 4, 5, 6 | syl2an2 673 | . . . 4 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → 𝐵 ∈ V) |
8 | simpr 477 | . . . 4 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V) | |
9 | fnsuppeq0 7654 | . . . 4 ⊢ (((𝐵 × {𝑍}) Fn 𝐵 ∧ 𝐵 ∈ V ∧ 𝑍 ∈ V) → (((𝐵 × {𝑍}) supp 𝑍) = ∅ ↔ (𝐵 × {𝑍}) = (𝐵 × {𝑍}))) | |
10 | 3, 7, 8, 9 | syl3anc 1351 | . . 3 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (((𝐵 × {𝑍}) supp 𝑍) = ∅ ↔ (𝐵 × {𝑍}) = (𝐵 × {𝑍}))) |
11 | 1, 10 | mpbird 249 | . 2 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → ((𝐵 × {𝑍}) supp 𝑍) = ∅) |
12 | supp0prc 7629 | . 2 ⊢ (¬ ((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → ((𝐵 × {𝑍}) supp 𝑍) = ∅) | |
13 | 11, 12 | pm2.61i 177 | 1 ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 Vcvv 3409 ∅c0 4173 {csn 4435 × cxp 5398 Fn wfn 6177 (class class class)co 6970 supp csupp 7626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-supp 7627 |
This theorem is referenced by: fczfsuppd 8638 cantnf 8942 mhp0cl 20033 |
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