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Mirrors > Home > ILE Home > Th. List > suppssof1 | GIF version |
Description: Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
Ref | Expression |
---|---|
suppssof1.s | ⊢ (φ → (◡A “ (V ∖ {𝑌})) ⊆ 𝐿) |
suppssof1.o | ⊢ ((φ ∧ v ∈ 𝑅) → (𝑌𝑂v) = 𝑍) |
suppssof1.a | ⊢ (φ → A:𝐷⟶𝑉) |
suppssof1.b | ⊢ (φ → B:𝐷⟶𝑅) |
suppssof1.d | ⊢ (φ → 𝐷 ∈ 𝑊) |
Ref | Expression |
---|---|
suppssof1 | ⊢ (φ → (◡(A ∘𝑓 𝑂B) “ (V ∖ {𝑍})) ⊆ 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppssof1.a | . . . . . 6 ⊢ (φ → A:𝐷⟶𝑉) | |
2 | ffn 4989 | . . . . . 6 ⊢ (A:𝐷⟶𝑉 → A Fn 𝐷) | |
3 | 1, 2 | syl 14 | . . . . 5 ⊢ (φ → A Fn 𝐷) |
4 | suppssof1.b | . . . . . 6 ⊢ (φ → B:𝐷⟶𝑅) | |
5 | ffn 4989 | . . . . . 6 ⊢ (B:𝐷⟶𝑅 → B Fn 𝐷) | |
6 | 4, 5 | syl 14 | . . . . 5 ⊢ (φ → B Fn 𝐷) |
7 | suppssof1.d | . . . . 5 ⊢ (φ → 𝐷 ∈ 𝑊) | |
8 | inidm 3140 | . . . . 5 ⊢ (𝐷 ∩ 𝐷) = 𝐷 | |
9 | eqidd 2038 | . . . . 5 ⊢ ((φ ∧ x ∈ 𝐷) → (A‘x) = (A‘x)) | |
10 | eqidd 2038 | . . . . 5 ⊢ ((φ ∧ x ∈ 𝐷) → (B‘x) = (B‘x)) | |
11 | 3, 6, 7, 7, 8, 9, 10 | offval 5661 | . . . 4 ⊢ (φ → (A ∘𝑓 𝑂B) = (x ∈ 𝐷 ↦ ((A‘x)𝑂(B‘x)))) |
12 | 11 | cnveqd 4454 | . . 3 ⊢ (φ → ◡(A ∘𝑓 𝑂B) = ◡(x ∈ 𝐷 ↦ ((A‘x)𝑂(B‘x)))) |
13 | 12 | imaeq1d 4610 | . 2 ⊢ (φ → (◡(A ∘𝑓 𝑂B) “ (V ∖ {𝑍})) = (◡(x ∈ 𝐷 ↦ ((A‘x)𝑂(B‘x))) “ (V ∖ {𝑍}))) |
14 | 1 | feqmptd 5169 | . . . . . 6 ⊢ (φ → A = (x ∈ 𝐷 ↦ (A‘x))) |
15 | 14 | cnveqd 4454 | . . . . 5 ⊢ (φ → ◡A = ◡(x ∈ 𝐷 ↦ (A‘x))) |
16 | 15 | imaeq1d 4610 | . . . 4 ⊢ (φ → (◡A “ (V ∖ {𝑌})) = (◡(x ∈ 𝐷 ↦ (A‘x)) “ (V ∖ {𝑌}))) |
17 | suppssof1.s | . . . 4 ⊢ (φ → (◡A “ (V ∖ {𝑌})) ⊆ 𝐿) | |
18 | 16, 17 | eqsstr3d 2974 | . . 3 ⊢ (φ → (◡(x ∈ 𝐷 ↦ (A‘x)) “ (V ∖ {𝑌})) ⊆ 𝐿) |
19 | suppssof1.o | . . 3 ⊢ ((φ ∧ v ∈ 𝑅) → (𝑌𝑂v) = 𝑍) | |
20 | funfvex 5135 | . . . . 5 ⊢ ((Fun A ∧ x ∈ dom A) → (A‘x) ∈ V) | |
21 | 20 | funfni 4942 | . . . 4 ⊢ ((A Fn 𝐷 ∧ x ∈ 𝐷) → (A‘x) ∈ V) |
22 | 3, 21 | sylan 267 | . . 3 ⊢ ((φ ∧ x ∈ 𝐷) → (A‘x) ∈ V) |
23 | 4 | ffvelrnda 5245 | . . 3 ⊢ ((φ ∧ x ∈ 𝐷) → (B‘x) ∈ 𝑅) |
24 | 18, 19, 22, 23 | suppssov1 5651 | . 2 ⊢ (φ → (◡(x ∈ 𝐷 ↦ ((A‘x)𝑂(B‘x))) “ (V ∖ {𝑍})) ⊆ 𝐿) |
25 | 13, 24 | eqsstrd 2973 | 1 ⊢ (φ → (◡(A ∘𝑓 𝑂B) “ (V ∖ {𝑍})) ⊆ 𝐿) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 Vcvv 2551 ∖ cdif 2908 ⊆ wss 2911 {csn 3367 ↦ cmpt 3809 ◡ccnv 4287 “ cima 4291 Fn wfn 4840 ⟶wf 4841 ‘cfv 4845 (class class class)co 5455 ∘𝑓 cof 5652 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-setind 4220 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-of 5654 |
This theorem is referenced by: (None) |
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