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Theorem filnetlem2 34117
 Description: Lemma for filnet 34120. 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 = 𝑛𝐹 ({𝑛} × 𝑛)
filnet.d 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}
Assertion
Ref Expression
filnetlem2 (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (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.
StepHypRef Expression
1 idref 6899 . . 3 (( I ↾ 𝐻) ⊆ 𝐷 ↔ ∀𝑧𝐻 𝑧𝐷𝑧)
2 ssid 3914 . . . . . 6 (1st𝑧) ⊆ (1st𝑧)
3 filnet.h . . . . . . 7 𝐻 = 𝑛𝐹 ({𝑛} × 𝑛)
4 filnet.d . . . . . . 7 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}
5 vex 3413 . . . . . . 7 𝑧 ∈ V
63, 4, 5, 5filnetlem1 34116 . . . . . 6 (𝑧𝐷𝑧 ↔ ((𝑧𝐻𝑧𝐻) ∧ (1st𝑧) ⊆ (1st𝑧)))
72, 6mpbiran2 709 . . . . 5 (𝑧𝐷𝑧 ↔ (𝑧𝐻𝑧𝐻))
87biimpri 231 . . . 4 ((𝑧𝐻𝑧𝐻) → 𝑧𝐷𝑧)
98anidms 570 . . 3 (𝑧𝐻𝑧𝐷𝑧)
101, 9mprgbir 3085 . 2 ( I ↾ 𝐻) ⊆ 𝐷
11 opabssxp 5612 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))} ⊆ (𝐻 × 𝐻)
124, 11eqsstri 3926 . 2 𝐷 ⊆ (𝐻 × 𝐻)
1310, 12pm3.2i 474 1 (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (𝐻 × 𝐻))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3858  {csn 4522  ∪ ciun 4883   class class class wbr 5032  {copab 5094   I cid 5429   × cxp 5522   ↾ cres 5526  ‘cfv 6335  1st c1st 7691 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343 This theorem is referenced by:  filnetlem3  34118  filnetlem4  34119
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