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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 5046 | . . . . . 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 5046 | . . . . . 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 3146 | . . . . 5 ⊢ (𝐷 ∩ 𝐷) = 𝐷 | |
9 | eqidd 2041 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) = (𝐴‘𝑥)) | |
10 | eqidd 2041 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) = (𝐵‘𝑥)) | |
11 | 3, 6, 7, 7, 8, 9, 10 | offval 5719 | . . . 4 ⊢ (𝜑 → (𝐴 ∘𝑓 𝑂𝐵) = (𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥)))) |
12 | 11 | cnveqd 4511 | . . 3 ⊢ (𝜑 → ◡(𝐴 ∘𝑓 𝑂𝐵) = ◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥)))) |
13 | 12 | imaeq1d 4667 | . 2 ⊢ (𝜑 → (◡(𝐴 ∘𝑓 𝑂𝐵) “ (V ∖ {𝑍})) = (◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) “ (V ∖ {𝑍}))) |
14 | 1 | feqmptd 5226 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥))) |
15 | 14 | cnveqd 4511 | . . . . 5 ⊢ (𝜑 → ◡𝐴 = ◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥))) |
16 | 15 | imaeq1d 4667 | . . . 4 ⊢ (𝜑 → (◡𝐴 “ (V ∖ {𝑌})) = (◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) “ (V ∖ {𝑌}))) |
17 | suppssof1.s | . . . 4 ⊢ (𝜑 → (◡𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿) | |
18 | 16, 17 | eqsstr3d 2980 | . . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) “ (V ∖ {𝑌})) ⊆ 𝐿) |
19 | suppssof1.o | . . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍) | |
20 | funfvex 5192 | . . . . 5 ⊢ ((Fun 𝐴 ∧ 𝑥 ∈ dom 𝐴) → (𝐴‘𝑥) ∈ V) | |
21 | 20 | funfni 4999 | . . . 4 ⊢ ((𝐴 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V) |
22 | 3, 21 | sylan 267 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V) |
23 | 4 | ffvelrnda 5302 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ∈ 𝑅) |
24 | 18, 19, 22, 23 | suppssov1 5709 | . 2 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) “ (V ∖ {𝑍})) ⊆ 𝐿) |
25 | 13, 24 | eqsstrd 2979 | 1 ⊢ (𝜑 → (◡(𝐴 ∘𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∖ cdif 2914 ⊆ wss 2917 {csn 3375 ↦ cmpt 3818 ◡ccnv 4344 “ cima 4348 Fn wfn 4897 ⟶wf 4898 ‘cfv 4902 (class class class)co 5512 ∘𝑓 cof 5710 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-of 5712 |
This theorem is referenced by: (None) |
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