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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 4401 | . . 3 ⊢ ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 supp 𝑍) = ∅) | |
| 2 | un0 4394 | . . . . . . . 8 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
| 3 | uncom 4158 | . . . . . . . 8 ⊢ (𝐴 ∪ ∅) = (∅ ∪ 𝐴) | |
| 4 | 2, 3 | eqtr3i 2767 | . . . . . . 7 ⊢ 𝐴 = (∅ ∪ 𝐴) |
| 5 | 4 | fneq2i 6666 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 Fn (∅ ∪ 𝐴)) |
| 6 | 5 | biimpi 216 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn (∅ ∪ 𝐴)) |
| 7 | 6 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝐹 Fn (∅ ∪ 𝐴)) |
| 8 | fnex 7237 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊) → 𝐹 ∈ V) | |
| 9 | 8 | 3adant3 1133 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝐹 ∈ V) |
| 10 | simp3 1139 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ 𝑉) | |
| 11 | 0in 4397 | . . . . 5 ⊢ (∅ ∩ 𝐴) = ∅ | |
| 12 | 11 | a1i 11 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (∅ ∩ 𝐴) = ∅) |
| 13 | fnsuppres 8216 | . . . 4 ⊢ ((𝐹 Fn (∅ ∪ 𝐴) ∧ (𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) ∧ (∅ ∩ 𝐴) = ∅) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍}))) | |
| 14 | 7, 9, 10, 12, 13 | syl121anc 1377 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍}))) |
| 15 | 1, 14 | bitr3id 285 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍}))) |
| 16 | fnresdm 6687 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
| 17 | 16 | 3ad2ant1 1134 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 ↾ 𝐴) = 𝐹) |
| 18 | 17 | eqeq1d 2739 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝐴) = (𝐴 × {𝑍}) ↔ 𝐹 = (𝐴 × {𝑍}))) |
| 19 | 15, 18 | bitrd 279 | 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 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 × cxp 5683 ↾ cres 5687 Fn wfn 6556 (class class class)co 7431 supp csupp 8185 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-supp 8186 |
| This theorem is referenced by: fczsupp0 8218 cantnf0 9715 mdegldg 26105 mdeg0 26109 suppovss 32690 fsuppind 42600 |
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