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Theorem filnetlem2 32711
Description: Lemma for filnet 32714. 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 6642 . . 3 (( I ↾ 𝐻) ⊆ 𝐷 ↔ ∀𝑧𝐻 𝑧𝐷𝑧)
2 ssid 3773 . . . . . 6 (1st𝑧) ⊆ (1st𝑧)
3 filnet.h . . . . . . 7 𝐻 = 𝑛𝐹 ({𝑛} × 𝑛)
4 filnet.d . . . . . . 7 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}
5 vex 3354 . . . . . . 7 𝑧 ∈ V
63, 4, 5, 5filnetlem1 32710 . . . . . 6 (𝑧𝐷𝑧 ↔ ((𝑧𝐻𝑧𝐻) ∧ (1st𝑧) ⊆ (1st𝑧)))
72, 6mpbiran2 689 . . . . 5 (𝑧𝐷𝑧 ↔ (𝑧𝐻𝑧𝐻))
87biimpri 218 . . . 4 ((𝑧𝐻𝑧𝐻) → 𝑧𝐷𝑧)
98anidms 556 . . 3 (𝑧𝐻𝑧𝐷𝑧)
101, 9mprgbir 3076 . 2 ( I ↾ 𝐻) ⊆ 𝐷
11 opabssxp 5333 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))} ⊆ (𝐻 × 𝐻)
124, 11eqsstri 3784 . 2 𝐷 ⊆ (𝐻 × 𝐻)
1310, 12pm3.2i 447 1 (( I ↾ 𝐻) ⊆ 𝐷𝐷 ⊆ (𝐻 × 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wcel 2145  wss 3723  {csn 4316   ciun 4654   class class class wbr 4786  {copab 4846   I cid 5156   × cxp 5247  cres 5251  cfv 6031  1st c1st 7313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039
This theorem is referenced by:  filnetlem3  32712  filnetlem4  32713
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