filnetlem2 - Mathbox for Jeff Hankins

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Theorem filnetlem2 31544

Description: Lemma for filnet 31547. The field of the direction. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

**Hypotheses**
Ref	Expression
filnet.h	⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)
filnet.d	⊢ 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1^st ‘𝑦) ⊆ (1^st ‘𝑥))}

**Assertion**
Ref	Expression
filnetlem2	⊢ (( I ↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻))

Distinct variable groups: 𝑥,𝑦,𝑛,𝐹 𝑥,𝐻,𝑦 Allowed substitution hints: 𝐷(𝑥,𝑦,𝑛) 𝐻(𝑛)

Proof of Theorem filnetlem2 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idref 6403	. . 3 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ↔ ∀𝑧 ∈ 𝐻 𝑧𝐷𝑧)
2		ssid 3587	. . . . . 6 ⊢ (1^st ‘𝑧) ⊆ (1^st ‘𝑧)
3		filnet.h	. . . . . . 7 ⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)
4		filnet.d	. . . . . . 7 ⊢ 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1^st ‘𝑦) ⊆ (1^st ‘𝑥))}
5		vex 3176	. . . . . . 7 ⊢ 𝑧 ∈ V
6	3, 4, 5, 5	filnetlem1 31543	. . . . . 6 ⊢ (𝑧𝐷𝑧 ↔ ((𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) ∧ (1^st ‘𝑧) ⊆ (1^st ‘𝑧)))
7	2, 6	mpbiran2 956	. . . . 5 ⊢ (𝑧𝐷𝑧 ↔ (𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻))
8	7	biimpri 217	. . . 4 ⊢ ((𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → 𝑧𝐷𝑧)
9	8	anidms 675	. . 3 ⊢ (𝑧 ∈ 𝐻 → 𝑧𝐷𝑧)
10	1, 9	mprgbir 2911	. 2 ⊢ ( I ↾ 𝐻) ⊆ 𝐷
11		opabssxp 5116	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1^st ‘𝑦) ⊆ (1^st ‘𝑥))} ⊆ (𝐻 × 𝐻)
12	4, 11	eqsstri 3598	. 2 ⊢ 𝐷 ⊆ (𝐻 × 𝐻)
13	10, 12	pm3.2i 470	1 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 {csn 4125 ∪ ciun 4455 class class class wbr 4583 {copab 4642 I cid 4948 × cxp 5036 ↾ cres 5040 ‘cfv 5804 1^st c1st 7057

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812

This theorem is referenced by: filnetlem3 31545 filnetlem4 31546

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