filnetlem2 - Mathbox for Jeff Hankins

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Theorem filnetlem2 36910

Description: Lemma for filnet 36913. 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 7142	. . 3 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ↔ ∀𝑧 ∈ 𝐻 𝑧𝐷𝑧)
2		ssid 3959	. . . . . 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 3459	. . . . . . 7 ⊢ 𝑧 ∈ V
6	3, 4, 5, 5	filnetlem1 36909	. . . . . 6 ⊢ (𝑧𝐷𝑧 ↔ ((𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) ∧ (1^st ‘𝑧) ⊆ (1^st ‘𝑧)))
7	2, 6	mpbiran2 722	. . . . 5 ⊢ (𝑧𝐷𝑧 ↔ (𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻))
8	7	biimpri 231	. . . 4 ⊢ ((𝑧 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) → 𝑧𝐷𝑧)
9	8	anidms 576	. . 3 ⊢ (𝑧 ∈ 𝐻 → 𝑧𝐷𝑧)
10	1, 9	mprgbir 3086	. 2 ⊢ ( I ↾ 𝐻) ⊆ 𝐷
11		opabssxp 5753	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1^st ‘𝑦) ⊆ (1^st ‘𝑥))} ⊆ (𝐻 × 𝐻)
12	4, 11	eqsstri 3983	. 2 ⊢ 𝐷 ⊆ (𝐻 × 𝐻)
13	10, 12	pm3.2i 475	1 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 400 = wceq 1570 ∈ wcel 2143 ⊆ wss 3905 {csn 4589 ∪ ciun 4956 class class class wbr 5109 {copab 5173 I cid 5555 × cxp 5659 ↾ cres 5663 ‘cfv 6536 1^st c1st 7980

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-pr 5404

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544

This theorem is referenced by: filnetlem3 36911 filnetlem4 36912

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