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Theorem filnetlem2 33285
Description: Lemma for filnet 33288. 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 6729 . . 3 (( I ↾ 𝐻) ⊆ 𝐷 ↔ ∀𝑧𝐻 𝑧𝐷𝑧)
2 ssid 3872 . . . . . 6 (1st𝑧) ⊆ (1st𝑧)
3 filnet.h . . . . . . 7 𝐻 = 𝑛𝐹 ({𝑛} × 𝑛)
4 filnet.d . . . . . . 7 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}
5 vex 3411 . . . . . . 7 𝑧 ∈ V
63, 4, 5, 5filnetlem1 33284 . . . . . 6 (𝑧𝐷𝑧 ↔ ((𝑧𝐻𝑧𝐻) ∧ (1st𝑧) ⊆ (1st𝑧)))
72, 6mpbiran2 698 . . . . 5 (𝑧𝐷𝑧 ↔ (𝑧𝐻𝑧𝐻))
87biimpri 220 . . . 4 ((𝑧𝐻𝑧𝐻) → 𝑧𝐷𝑧)
98anidms 559 . . 3 (𝑧𝐻𝑧𝐷𝑧)
101, 9mprgbir 3096 . 2 ( I ↾ 𝐻) ⊆ 𝐷
11 opabssxp 5489 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))} ⊆ (𝐻 × 𝐻)
124, 11eqsstri 3884 . 2 𝐷 ⊆ (𝐻 × 𝐻)
1310, 12pm3.2i 463 1 (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (𝐻 × 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wa 387   = wceq 1508  wcel 2051  wss 3822  {csn 4435   ciun 4788   class class class wbr 4925  {copab 4987   I cid 5307   × cxp 5401  cres 5405  cfv 6185  1st c1st 7497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193
This theorem is referenced by:  filnetlem3  33286  filnetlem4  33287
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