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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 5332 | . . . . . 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 5332 | . . . . . 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 3327 | . . . . 5 ⊢ (𝐷 ∩ 𝐷) = 𝐷 | |
9 | eqidd 2165 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) = (𝐴‘𝑥)) | |
10 | eqidd 2165 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) = (𝐵‘𝑥)) | |
11 | 3, 6, 7, 7, 8, 9, 10 | offval 6052 | . . . 4 ⊢ (𝜑 → (𝐴 ∘𝑓 𝑂𝐵) = (𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥)))) |
12 | 11 | cnveqd 4775 | . . 3 ⊢ (𝜑 → ◡(𝐴 ∘𝑓 𝑂𝐵) = ◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥)))) |
13 | 12 | imaeq1d 4940 | . 2 ⊢ (𝜑 → (◡(𝐴 ∘𝑓 𝑂𝐵) “ (V ∖ {𝑍})) = (◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) “ (V ∖ {𝑍}))) |
14 | 1 | feqmptd 5534 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥))) |
15 | 14 | cnveqd 4775 | . . . . 5 ⊢ (𝜑 → ◡𝐴 = ◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥))) |
16 | 15 | imaeq1d 4940 | . . . 4 ⊢ (𝜑 → (◡𝐴 “ (V ∖ {𝑌})) = (◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) “ (V ∖ {𝑌}))) |
17 | suppssof1.s | . . . 4 ⊢ (𝜑 → (◡𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿) | |
18 | 16, 17 | eqsstrrd 3175 | . . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ (𝐴‘𝑥)) “ (V ∖ {𝑌})) ⊆ 𝐿) |
19 | suppssof1.o | . . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍) | |
20 | funfvex 5498 | . . . . 5 ⊢ ((Fun 𝐴 ∧ 𝑥 ∈ dom 𝐴) → (𝐴‘𝑥) ∈ V) | |
21 | 20 | funfni 5283 | . . . 4 ⊢ ((𝐴 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V) |
22 | 3, 21 | sylan 281 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ∈ V) |
23 | 4 | ffvelrnda 5615 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ∈ 𝑅) |
24 | 18, 19, 22, 23 | suppssov1 6042 | . 2 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ ((𝐴‘𝑥)𝑂(𝐵‘𝑥))) “ (V ∖ {𝑍})) ⊆ 𝐿) |
25 | 13, 24 | eqsstrd 3174 | 1 ⊢ (𝜑 → (◡(𝐴 ∘𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 Vcvv 2722 ∖ cdif 3109 ⊆ wss 3112 {csn 3571 ↦ cmpt 4038 ◡ccnv 4598 “ cima 4602 Fn wfn 5178 ⟶wf 5179 ‘cfv 5183 (class class class)co 5837 ∘𝑓 cof 6043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-setind 4509 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-ov 5840 df-oprab 5841 df-mpo 5842 df-of 6045 |
This theorem is referenced by: (None) |
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