filnetlem2 - Mathbox for Jeff Hankins

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Theorem filnetlem2 36779

Description: Lemma for filnet 36782. 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 7143	. . 3 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ↔ ∀𝑧 ∈ 𝐻 𝑧𝐷𝑧)
2		ssid 3967	. . . . . 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 3467	. . . . . . 7 ⊢ 𝑧 ∈ V
6	3, 4, 5, 5	filnetlem1 36778	. . . . . 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 3092	. 2 ⊢ ( I ↾ 𝐻) ⊆ 𝐷
11		opabssxp 5754	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1^st ‘𝑦) ⊆ (1^st ‘𝑥))} ⊆ (𝐻 × 𝐻)
12	4, 11	eqsstri 3991	. 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 1567 ∈ wcel 2149 ⊆ wss 3913 {csn 4594 ∪ ciun 4960 class class class wbr 5113 {copab 5177 I cid 5556 × cxp 5660 ↾ cres 5664 ‘cfv 6537 1^st c1st 7984

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-pr 5405

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545

This theorem is referenced by: filnetlem3 36780 filnetlem4 36781

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