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Mirrors > Home > MPE Home > Th. List > fvn0elsuppb | Structured version Visualization version GIF version |
Description: The function value for a given argument is not empty iff the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 4-Apr-2020.) |
Ref | Expression |
---|---|
fvn0elsuppb | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝐺 Fn 𝐵) → ((𝐺‘𝑋) ≠ ∅ ↔ 𝑋 ∈ (𝐺 supp ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvn0elsupp 7840 | . . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺‘𝑋) ≠ ∅)) → 𝑋 ∈ (𝐺 supp ∅)) | |
2 | 1 | exp43 439 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑋 ∈ 𝐵 → (𝐺 Fn 𝐵 → ((𝐺‘𝑋) ≠ ∅ → 𝑋 ∈ (𝐺 supp ∅))))) |
3 | 2 | 3imp 1107 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝐺 Fn 𝐵) → ((𝐺‘𝑋) ≠ ∅ → 𝑋 ∈ (𝐺 supp ∅))) |
4 | simp3 1134 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝐺 Fn 𝐵) → 𝐺 Fn 𝐵) | |
5 | simp1 1132 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝐺 Fn 𝐵) → 𝐵 ∈ 𝑉) | |
6 | 0ex 5203 | . . . . 5 ⊢ ∅ ∈ V | |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝐺 Fn 𝐵) → ∅ ∈ V) |
8 | elsuppfn 7832 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅))) | |
9 | 4, 5, 7, 8 | syl3anc 1367 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝐺 Fn 𝐵) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅))) |
10 | simpr 487 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅) → (𝐺‘𝑋) ≠ ∅) | |
11 | 9, 10 | syl6bi 255 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝐺 Fn 𝐵) → (𝑋 ∈ (𝐺 supp ∅) → (𝐺‘𝑋) ≠ ∅)) |
12 | 3, 11 | impbid 214 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝐺 Fn 𝐵) → ((𝐺‘𝑋) ≠ ∅ ↔ 𝑋 ∈ (𝐺 supp ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∅c0 4290 Fn wfn 6344 ‘cfv 6349 (class class class)co 7150 supp csupp 7824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-supp 7825 |
This theorem is referenced by: brcic 17062 |
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