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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 2611 | . . 3 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) = (𝐵 × {𝑍})) | |
2 | fnconstg 6006 | . . . . 5 ⊢ (𝑍 ∈ V → (𝐵 × {𝑍}) Fn 𝐵) | |
3 | 2 | adantl 481 | . . . 4 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) Fn 𝐵) |
4 | snnzg 4251 | . . . . . 6 ⊢ (𝑍 ∈ V → {𝑍} ≠ ∅) | |
5 | 4 | adantl 481 | . . . . 5 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → {𝑍} ≠ ∅) |
6 | simpl 472 | . . . . 5 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (𝐵 × {𝑍}) ∈ V) | |
7 | xpexcnv 7001 | . . . . 5 ⊢ (({𝑍} ≠ ∅ ∧ (𝐵 × {𝑍}) ∈ V) → 𝐵 ∈ V) | |
8 | 5, 6, 7 | syl2anc 691 | . . . 4 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → 𝐵 ∈ V) |
9 | simpr 476 | . . . 4 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V) | |
10 | fnsuppeq0 7210 | . . . 4 ⊢ (((𝐵 × {𝑍}) Fn 𝐵 ∧ 𝐵 ∈ V ∧ 𝑍 ∈ V) → (((𝐵 × {𝑍}) supp 𝑍) = ∅ ↔ (𝐵 × {𝑍}) = (𝐵 × {𝑍}))) | |
11 | 3, 8, 9, 10 | syl3anc 1318 | . . 3 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → (((𝐵 × {𝑍}) supp 𝑍) = ∅ ↔ (𝐵 × {𝑍}) = (𝐵 × {𝑍}))) |
12 | 1, 11 | mpbird 246 | . 2 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → ((𝐵 × {𝑍}) supp 𝑍) = ∅) |
13 | supp0prc 7185 | . 2 ⊢ (¬ ((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ V) → ((𝐵 × {𝑍}) supp 𝑍) = ∅) | |
14 | 12, 13 | pm2.61i 175 | 1 ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∅c0 3874 {csn 4125 × cxp 5036 Fn wfn 5799 (class class class)co 6549 supp csupp 7182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-supp 7183 |
This theorem is referenced by: fczfsuppd 8176 cantnf 8473 |
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