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Mirrors > Home > MPE Home > Th. List > Mathboxes > filnetlem2 | Structured version Visualization version GIF version |
Description: Lemma for filnet 33627. 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 6900 | . . 3 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ↔ ∀𝑧 ∈ 𝐻 𝑧𝐷𝑧) | |
2 | ssid 3986 | . . . . . 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 3495 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
6 | 3, 4, 5, 5 | filnetlem1 33623 | . . . . . 6 ⊢ (𝑧𝐷𝑧 ↔ ((𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) ∧ (1st ‘𝑧) ⊆ (1st ‘𝑧))) |
7 | 2, 6 | mpbiran2 706 | . . . . 5 ⊢ (𝑧𝐷𝑧 ↔ (𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) |
8 | 7 | biimpri 229 | . . . 4 ⊢ ((𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → 𝑧𝐷𝑧) |
9 | 8 | anidms 567 | . . 3 ⊢ (𝑧 ∈ 𝐻 → 𝑧𝐷𝑧) |
10 | 1, 9 | mprgbir 3150 | . 2 ⊢ ( I ↾ 𝐻) ⊆ 𝐷 |
11 | opabssxp 5636 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))} ⊆ (𝐻 × 𝐻) | |
12 | 4, 11 | eqsstri 3998 | . 2 ⊢ 𝐷 ⊆ (𝐻 × 𝐻) |
13 | 10, 12 | pm3.2i 471 | 1 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 {csn 4557 ∪ ciun 4910 class class class wbr 5057 {copab 5119 I cid 5452 × cxp 5546 ↾ cres 5550 ‘cfv 6348 1st c1st 7676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 |
This theorem is referenced by: filnetlem3 33625 filnetlem4 33626 |
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