Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > suppssrg | Structured version Visualization version GIF version |
Description: A function is zero outside its support. Version of suppssr 7876 avoiding ax-rep 5160 by assuming 𝐹 is a set rather than its domain 𝐴. (Contributed by SN, 5-May-2024.) |
Ref | Expression |
---|---|
suppssrg.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
suppssrg.n | ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) |
suppssrg.a | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
suppssrg.z | ⊢ (𝜑 → 𝑍 ∈ 𝑈) |
Ref | Expression |
---|---|
suppssrg | ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3870 | . 2 ⊢ (𝑋 ∈ (𝐴 ∖ 𝑊) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊)) | |
2 | suppssrg.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
3 | 2 | ffnd 6504 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
4 | suppssrg.a | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
5 | suppssrg.z | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
6 | elsuppfng 7850 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍))) | |
7 | 3, 4, 5, 6 | syl3anc 1368 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍))) |
8 | suppssrg.n | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) | |
9 | 8 | sseld 3893 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) → 𝑋 ∈ 𝑊)) |
10 | 7, 9 | sylbird 263 | . . . . 5 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍) → 𝑋 ∈ 𝑊)) |
11 | 10 | expdimp 456 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 → 𝑋 ∈ 𝑊)) |
12 | 11 | necon1bd 2969 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ 𝑋 ∈ 𝑊 → (𝐹‘𝑋) = 𝑍)) |
13 | 12 | impr 458 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊)) → (𝐹‘𝑋) = 𝑍) |
14 | 1, 13 | sylan2b 596 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 ∖ cdif 3857 ⊆ wss 3860 Fn wfn 6335 ⟶wf 6336 ‘cfv 6340 (class class class)co 7156 supp csupp 7841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5037 df-opab 5099 df-id 5434 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-fv 6348 df-ov 7159 df-oprab 7160 df-mpo 7161 df-supp 7842 |
This theorem is referenced by: psrbaglesupp 20699 psrbaglefi 20707 |
Copyright terms: Public domain | W3C validator |