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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 2825 | . . 3 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) = (𝐵 × {𝑍})) | |
2 | fnconstg 6570 | . . . . 5 ⊢ (𝑍 ∈ V → (𝐵 × {𝑍}) Fn 𝐵) | |
3 | 2 | adantl 484 | . . . 4 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) Fn 𝐵) |
4 | snnzg 4713 | . . . . 5 ⊢ (𝑍 ∈ V → {𝑍} ≠ ∅) | |
5 | simpl 485 | . . . . 5 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) ∈ V) | |
6 | xpexcnv 7628 | . . . . 5 ⊢ (({𝑍} ≠ ∅ ∧ (𝐵 × {𝑍}) ∈ V) → 𝐵 ∈ V) | |
7 | 4, 5, 6 | syl2an2 684 | . . . 4 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → 𝐵 ∈ V) |
8 | simpr 487 | . . . 4 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V) | |
9 | fnsuppeq0 7861 | . . . 4 ⊢ (((𝐵 × {𝑍}) Fn 𝐵 ∧ 𝐵 ∈ V ∧ 𝑍 ∈ V) → (((𝐵 × {𝑍}) supp 𝑍) = ∅ ↔ (𝐵 × {𝑍}) = (𝐵 × {𝑍}))) | |
10 | 3, 7, 8, 9 | syl3anc 1367 | . . 3 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (((𝐵 × {𝑍}) supp 𝑍) = ∅ ↔ (𝐵 × {𝑍}) = (𝐵 × {𝑍}))) |
11 | 1, 10 | mpbird 259 | . 2 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → ((𝐵 × {𝑍}) supp 𝑍) = ∅) |
12 | supp0prc 7836 | . 2 ⊢ (¬ ((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → ((𝐵 × {𝑍}) supp 𝑍) = ∅) | |
13 | 11, 12 | pm2.61i 184 | 1 ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 Vcvv 3497 ∅c0 4294 {csn 4570 × cxp 5556 Fn wfn 6353 (class class class)co 7159 supp csupp 7833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-supp 7834 |
This theorem is referenced by: fczfsuppd 8854 cantnf 9159 mhp0cl 20340 |
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