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Mirrors > Home > MPE Home > Th. List > Mathboxes > filnetlem2 | Structured version Visualization version GIF version |
Description: Lemma for filnet 33288. 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 6729 | . . 3 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ↔ ∀𝑧 ∈ 𝐻 𝑧𝐷𝑧) | |
2 | ssid 3872 | . . . . . 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 3411 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
6 | 3, 4, 5, 5 | filnetlem1 33284 | . . . . . 6 ⊢ (𝑧𝐷𝑧 ↔ ((𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) ∧ (1st ‘𝑧) ⊆ (1st ‘𝑧))) |
7 | 2, 6 | mpbiran2 698 | . . . . 5 ⊢ (𝑧𝐷𝑧 ↔ (𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) |
8 | 7 | biimpri 220 | . . . 4 ⊢ ((𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → 𝑧𝐷𝑧) |
9 | 8 | anidms 559 | . . 3 ⊢ (𝑧 ∈ 𝐻 → 𝑧𝐷𝑧) |
10 | 1, 9 | mprgbir 3096 | . 2 ⊢ ( I ↾ 𝐻) ⊆ 𝐷 |
11 | opabssxp 5489 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))} ⊆ (𝐻 × 𝐻) | |
12 | 4, 11 | eqsstri 3884 | . 2 ⊢ 𝐷 ⊆ (𝐻 × 𝐻) |
13 | 10, 12 | pm3.2i 463 | 1 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 = wceq 1508 ∈ wcel 2051 ⊆ wss 3822 {csn 4435 ∪ ciun 4788 class class class wbr 4925 {copab 4987 I cid 5307 × cxp 5401 ↾ cres 5405 ‘cfv 6185 1st c1st 7497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 |
This theorem is referenced by: filnetlem3 33286 filnetlem4 33287 |
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