filnetlem2 - Mathbox for Jeff Hankins

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Theorem filnetlem2 37167

Description: Lemma for filnet 37170. 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 7149	. . 3 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ↔ ∀𝑧 ∈ 𝐻 𝑧𝐷𝑧)
2		ssid 3953	. . . . . 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 3455	. . . . . . 7 ⊢ 𝑧 ∈ V
6	3, 4, 5, 5	filnetlem1 37166	. . . . . 6 ⊢ (𝑧𝐷𝑧 ↔ ((𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) ∧ (1^st ‘𝑧) ⊆ (1^st ‘𝑧)))
7	2, 6	mpbiran2 723	. . . . 5 ⊢ (𝑧𝐷𝑧 ↔ (𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻))
8	7	biimpri 231	. . . 4 ⊢ ((𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → 𝑧𝐷𝑧)
9	8	anidms 577	. . 3 ⊢ (𝑧 ∈ 𝐻 → 𝑧𝐷𝑧)
10	1, 9	mprgbir 3084	. 2 ⊢ ( I ↾ 𝐻) ⊆ 𝐷
11		opabssxp 5743	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1^st ‘𝑦) ⊆ (1^st ‘𝑥))} ⊆ (𝐻 × 𝐻)
12	4, 11	eqsstri 3977	. 2 ⊢ 𝐷 ⊆ (𝐻 × 𝐻)
13	10, 12	pm3.2i 476	1 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⊆ wss 3899 {csn 4584 ∪ ciun 4951 class class class wbr 5103 {copab 5167 I cid 5545 × cxp 5649 ↾ cres 5653 ‘cfv 6538 1^st c1st 7999

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-pr 5391

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546

This theorem is used by: filnetlem3 37168 filnetlem4 37169

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