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Mirrors > Home > MPE Home > Th. List > fnsuppeq0 | Structured version Visualization version GIF version |
Description: The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 28-May-2019.) |
Ref | Expression |
---|---|
fnsuppeq0 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss0b 4351 | . . 3 ⊢ ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 supp 𝑍) = ∅) | |
2 | un0 4344 | . . . . . . . 8 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
3 | uncom 4129 | . . . . . . . 8 ⊢ (𝐴 ∪ ∅) = (∅ ∪ 𝐴) | |
4 | 2, 3 | eqtr3i 2846 | . . . . . . 7 ⊢ 𝐴 = (∅ ∪ 𝐴) |
5 | 4 | fneq2i 6451 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 Fn (∅ ∪ 𝐴)) |
6 | 5 | biimpi 218 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn (∅ ∪ 𝐴)) |
7 | 6 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝐹 Fn (∅ ∪ 𝐴)) |
8 | fnex 6980 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊) → 𝐹 ∈ V) | |
9 | 8 | 3adant3 1128 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝐹 ∈ V) |
10 | simp3 1134 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ 𝑉) | |
11 | 0in 4347 | . . . . 5 ⊢ (∅ ∩ 𝐴) = ∅ | |
12 | 11 | a1i 11 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (∅ ∩ 𝐴) = ∅) |
13 | fnsuppres 7857 | . . . 4 ⊢ ((𝐹 Fn (∅ ∪ 𝐴) ∧ (𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) ∧ (∅ ∩ 𝐴) = ∅) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍}))) | |
14 | 7, 9, 10, 12, 13 | syl121anc 1371 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍}))) |
15 | 1, 14 | syl5bbr 287 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍}))) |
16 | fnresdm 6466 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
17 | 16 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 ↾ 𝐴) = 𝐹) |
18 | 17 | eqeq1d 2823 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝐴) = (𝐴 × {𝑍}) ↔ 𝐹 = (𝐴 × {𝑍}))) |
19 | 15, 18 | bitrd 281 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {csn 4567 × cxp 5553 ↾ cres 5557 Fn wfn 6350 (class class class)co 7156 supp csupp 7830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-supp 7831 |
This theorem is referenced by: fczsupp0 7859 cantnf0 9138 mdegldg 24660 mdeg0 24664 suppovss 30426 |
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