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Mirrors > Home > MPE Home > Th. List > 00lsp | Structured version Visualization version GIF version |
Description: fvco4i 6762 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
Ref | Expression |
---|---|
00lsp | ⊢ ∅ = (LSpan‘∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5211 | . . 3 ⊢ ∅ ∈ V | |
2 | base0 16536 | . . . 4 ⊢ ∅ = (Base‘∅) | |
3 | 00lss 19713 | . . . 4 ⊢ ∅ = (LSubSp‘∅) | |
4 | eqid 2821 | . . . 4 ⊢ (LSpan‘∅) = (LSpan‘∅) | |
5 | 2, 3, 4 | lspfval 19745 | . . 3 ⊢ (∅ ∈ V → (LSpan‘∅) = (𝑎 ∈ 𝒫 ∅ ↦ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏})) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ (LSpan‘∅) = (𝑎 ∈ 𝒫 ∅ ↦ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏}) |
7 | eqid 2821 | . . . . 5 ⊢ (𝑎 ∈ 𝒫 ∅ ↦ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏}) = (𝑎 ∈ 𝒫 ∅ ↦ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏}) | |
8 | 7 | dmmpt 6094 | . . . 4 ⊢ dom (𝑎 ∈ 𝒫 ∅ ↦ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏}) = {𝑎 ∈ 𝒫 ∅ ∣ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏} ∈ V} |
9 | rab0 4337 | . . . . . . . . 9 ⊢ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏} = ∅ | |
10 | 9 | inteqi 4880 | . . . . . . . 8 ⊢ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏} = ∩ ∅ |
11 | int0 4890 | . . . . . . . 8 ⊢ ∩ ∅ = V | |
12 | 10, 11 | eqtri 2844 | . . . . . . 7 ⊢ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏} = V |
13 | vprc 5219 | . . . . . . 7 ⊢ ¬ V ∈ V | |
14 | 12, 13 | eqneltri 2906 | . . . . . 6 ⊢ ¬ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏} ∈ V |
15 | 14 | rgenw 3150 | . . . . 5 ⊢ ∀𝑎 ∈ 𝒫 ∅ ¬ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏} ∈ V |
16 | rabeq0 4338 | . . . . 5 ⊢ ({𝑎 ∈ 𝒫 ∅ ∣ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏} ∈ V} = ∅ ↔ ∀𝑎 ∈ 𝒫 ∅ ¬ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏} ∈ V) | |
17 | 15, 16 | mpbir 233 | . . . 4 ⊢ {𝑎 ∈ 𝒫 ∅ ∣ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏} ∈ V} = ∅ |
18 | 8, 17 | eqtri 2844 | . . 3 ⊢ dom (𝑎 ∈ 𝒫 ∅ ↦ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏}) = ∅ |
19 | mptrel 5697 | . . . 4 ⊢ Rel (𝑎 ∈ 𝒫 ∅ ↦ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏}) | |
20 | reldm0 5798 | . . . 4 ⊢ (Rel (𝑎 ∈ 𝒫 ∅ ↦ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏}) → ((𝑎 ∈ 𝒫 ∅ ↦ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏}) = ∅ ↔ dom (𝑎 ∈ 𝒫 ∅ ↦ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏}) = ∅)) | |
21 | 19, 20 | ax-mp 5 | . . 3 ⊢ ((𝑎 ∈ 𝒫 ∅ ↦ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏}) = ∅ ↔ dom (𝑎 ∈ 𝒫 ∅ ↦ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏}) = ∅) |
22 | 18, 21 | mpbir 233 | . 2 ⊢ (𝑎 ∈ 𝒫 ∅ ↦ ∩ {𝑏 ∈ ∅ ∣ 𝑎 ⊆ 𝑏}) = ∅ |
23 | 6, 22 | eqtr2i 2845 | 1 ⊢ ∅ = (LSpan‘∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 ∩ cint 4876 ↦ cmpt 5146 dom cdm 5555 Rel wrel 5560 ‘cfv 6355 LSpanclspn 19743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-slot 16487 df-base 16489 df-lss 19704 df-lsp 19744 |
This theorem is referenced by: rspval 19965 |
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