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Theorem filnetlem1 31543
 Description: Lemma for filnet 31547. Change variables. (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))}
filnetlem1.a 𝐴 ∈ V
filnetlem1.b 𝐵 ∈ V
Assertion
Ref Expression
filnetlem1 (𝐴𝐷𝐵 ↔ ((𝐴𝐻𝐵𝐻) ∧ (1st𝐵) ⊆ (1st𝐴)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑛,𝑦,𝐹   𝑥,𝐻,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐴(𝑛)   𝐵(𝑛)   𝐷(𝑥,𝑦,𝑛)   𝐻(𝑛)

Proof of Theorem filnetlem1
StepHypRef Expression
1 fveq2 6103 . . . 4 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
21sseq2d 3596 . . 3 (𝑥 = 𝐴 → ((1st𝑦) ⊆ (1st𝑥) ↔ (1st𝑦) ⊆ (1st𝐴)))
3 fveq2 6103 . . . 4 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
43sseq1d 3595 . . 3 (𝑦 = 𝐵 → ((1st𝑦) ⊆ (1st𝐴) ↔ (1st𝐵) ⊆ (1st𝐴)))
52, 4sylan9bb 732 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → ((1st𝑦) ⊆ (1st𝑥) ↔ (1st𝐵) ⊆ (1st𝐴)))
6 filnet.d . 2 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}
75, 6brab2ga 5117 1 (𝐴𝐷𝐵 ↔ ((𝐴𝐻𝐵𝐻) ∧ (1st𝐵) ⊆ (1st𝐴)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  {csn 4125  ∪ ciun 4455   class class class wbr 4583  {copab 4642   × cxp 5036  ‘cfv 5804  1st c1st 7057 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-iota 5768  df-fv 5812 This theorem is referenced by:  filnetlem2  31544  filnetlem3  31545  filnetlem4  31546
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