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Theorem filnetlem1 34781
Description: Lemma for filnet 34785. 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 6839 . . . 4 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
21sseq2d 3974 . . 3 (𝑥 = 𝐴 → ((1st𝑦) ⊆ (1st𝑥) ↔ (1st𝑦) ⊆ (1st𝐴)))
3 fveq2 6839 . . . 4 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
43sseq1d 3973 . . 3 (𝑦 = 𝐵 → ((1st𝑦) ⊆ (1st𝐴) ↔ (1st𝐵) ⊆ (1st𝐴)))
52, 4sylan9bb 510 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → ((1st𝑦) ⊆ (1st𝑥) ↔ (1st𝐵) ⊆ (1st𝐴)))
6 filnet.d . 2 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}
75, 6brab2a 5723 1 (𝐴𝐷𝐵 ↔ ((𝐴𝐻𝐵𝐻) ∧ (1st𝐵) ⊆ (1st𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3443  wss 3908  {csn 4584   ciun 4952   class class class wbr 5103  {copab 5165   × cxp 5629  cfv 6493  1st c1st 7911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-xp 5637  df-iota 6445  df-fv 6501
This theorem is referenced by:  filnetlem2  34782  filnetlem3  34783  filnetlem4  34784
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