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Mirrors > Home > MPE Home > Th. List > Mathboxes > filnetlem2 | Structured version Visualization version GIF version |
Description: Lemma for filnet 33732. The field of the direction. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.) |
Ref | Expression |
---|---|
filnet.h | ⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) |
filnet.d | ⊢ 𝐷 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))} |
Ref | Expression |
---|---|
filnetlem2 | ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idref 6910 | . . 3 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ↔ ∀𝑧 ∈ 𝐻 𝑧𝐷𝑧) | |
2 | ssid 3991 | . . . . . 6 ⊢ (1st ‘𝑧) ⊆ (1st ‘𝑧) | |
3 | filnet.h | . . . . . . 7 ⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) | |
4 | filnet.d | . . . . . . 7 ⊢ 𝐷 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))} | |
5 | vex 3499 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
6 | 3, 4, 5, 5 | filnetlem1 33728 | . . . . . 6 ⊢ (𝑧𝐷𝑧 ↔ ((𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) ∧ (1st ‘𝑧) ⊆ (1st ‘𝑧))) |
7 | 2, 6 | mpbiran2 708 | . . . . 5 ⊢ (𝑧𝐷𝑧 ↔ (𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) |
8 | 7 | biimpri 230 | . . . 4 ⊢ ((𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → 𝑧𝐷𝑧) |
9 | 8 | anidms 569 | . . 3 ⊢ (𝑧 ∈ 𝐻 → 𝑧𝐷𝑧) |
10 | 1, 9 | mprgbir 3155 | . 2 ⊢ ( I ↾ 𝐻) ⊆ 𝐷 |
11 | opabssxp 5645 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))} ⊆ (𝐻 × 𝐻) | |
12 | 4, 11 | eqsstri 4003 | . 2 ⊢ 𝐷 ⊆ (𝐻 × 𝐻) |
13 | 10, 12 | pm3.2i 473 | 1 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 {csn 4569 ∪ ciun 4921 class class class wbr 5068 {copab 5130 I cid 5461 × cxp 5555 ↾ cres 5559 ‘cfv 6357 1st c1st 7689 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 |
This theorem is referenced by: filnetlem3 33730 filnetlem4 33731 |
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