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Theorem filnetlem1 34304
Description: Lemma for filnet 34308. 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 6717 . . . 4 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
21sseq2d 3933 . . 3 (𝑥 = 𝐴 → ((1st𝑦) ⊆ (1st𝑥) ↔ (1st𝑦) ⊆ (1st𝐴)))
3 fveq2 6717 . . . 4 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
43sseq1d 3932 . . 3 (𝑦 = 𝐵 → ((1st𝑦) ⊆ (1st𝐴) ↔ (1st𝐵) ⊆ (1st𝐴)))
52, 4sylan9bb 513 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → ((1st𝑦) ⊆ (1st𝑥) ↔ (1st𝐵) ⊆ (1st𝐴)))
6 filnet.d . 2 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}
75, 6brab2a 5641 1 (𝐴𝐷𝐵 ↔ ((𝐴𝐻𝐵𝐻) ∧ (1st𝐵) ⊆ (1st𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wcel 2110  Vcvv 3408  wss 3866  {csn 4541   ciun 4904   class class class wbr 5053  {copab 5115   × cxp 5549  cfv 6380  1st c1st 7759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-xp 5557  df-iota 6338  df-fv 6388
This theorem is referenced by:  filnetlem2  34305  filnetlem3  34306  filnetlem4  34307
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