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Theorem filnetlem2 35772
Description: Lemma for filnet 35775. 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 7140 . . 3 (( I ↾ 𝐻) ⊆ 𝐷 ↔ ∀𝑧𝐻 𝑧𝐷𝑧)
2 ssid 3999 . . . . . 6 (1st𝑧) ⊆ (1st𝑧)
3 filnet.h . . . . . . 7 𝐻 = 𝑛𝐹 ({𝑛} × 𝑛)
4 filnet.d . . . . . . 7 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}
5 vex 3472 . . . . . . 7 𝑧 ∈ V
63, 4, 5, 5filnetlem1 35771 . . . . . 6 (𝑧𝐷𝑧 ↔ ((𝑧𝐻𝑧𝐻) ∧ (1st𝑧) ⊆ (1st𝑧)))
72, 6mpbiran2 707 . . . . 5 (𝑧𝐷𝑧 ↔ (𝑧𝐻𝑧𝐻))
87biimpri 227 . . . 4 ((𝑧𝐻𝑧𝐻) → 𝑧𝐷𝑧)
98anidms 566 . . 3 (𝑧𝐻𝑧𝐷𝑧)
101, 9mprgbir 3062 . 2 ( I ↾ 𝐻) ⊆ 𝐷
11 opabssxp 5761 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))} ⊆ (𝐻 × 𝐻)
124, 11eqsstri 4011 . 2 𝐷 ⊆ (𝐻 × 𝐻)
1310, 12pm3.2i 470 1 (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (𝐻 × 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1533  wcel 2098  wss 3943  {csn 4623   ciun 4990   class class class wbr 5141  {copab 5203   I cid 5566   × cxp 5667  cres 5671  cfv 6537  1st c1st 7972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  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  35773  filnetlem4  35774
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