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Theorem filnetlem2 33624
Description: Lemma for filnet 33627. 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 6900 . . 3 (( I ↾ 𝐻) ⊆ 𝐷 ↔ ∀𝑧𝐻 𝑧𝐷𝑧)
2 ssid 3986 . . . . . 6 (1st𝑧) ⊆ (1st𝑧)
3 filnet.h . . . . . . 7 𝐻 = 𝑛𝐹 ({𝑛} × 𝑛)
4 filnet.d . . . . . . 7 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}
5 vex 3495 . . . . . . 7 𝑧 ∈ V
63, 4, 5, 5filnetlem1 33623 . . . . . 6 (𝑧𝐷𝑧 ↔ ((𝑧𝐻𝑧𝐻) ∧ (1st𝑧) ⊆ (1st𝑧)))
72, 6mpbiran2 706 . . . . 5 (𝑧𝐷𝑧 ↔ (𝑧𝐻𝑧𝐻))
87biimpri 229 . . . 4 ((𝑧𝐻𝑧𝐻) → 𝑧𝐷𝑧)
98anidms 567 . . 3 (𝑧𝐻𝑧𝐷𝑧)
101, 9mprgbir 3150 . 2 ( I ↾ 𝐻) ⊆ 𝐷
11 opabssxp 5636 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))} ⊆ (𝐻 × 𝐻)
124, 11eqsstri 3998 . 2 𝐷 ⊆ (𝐻 × 𝐻)
1310, 12pm3.2i 471 1 (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (𝐻 × 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wcel 2105  wss 3933  {csn 4557   ciun 4910   class class class wbr 5057  {copab 5119   I cid 5452   × cxp 5546  cres 5550  cfv 6348  1st c1st 7676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  filnetlem3  33625  filnetlem4  33626
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