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Theorem filnetlem1 33623
Description: Lemma for filnet 33627. 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 6663 . . . 4 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
21sseq2d 3996 . . 3 (𝑥 = 𝐴 → ((1st𝑦) ⊆ (1st𝑥) ↔ (1st𝑦) ⊆ (1st𝐴)))
3 fveq2 6663 . . . 4 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
43sseq1d 3995 . . 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 5637 1 (𝐴𝐷𝐵 ↔ ((𝐴𝐻𝐵𝐻) ∧ (1st𝐵) ⊆ (1st𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  {csn 4557   ciun 4910   class class class wbr 5057  {copab 5119   × cxp 5546  cfv 6348  1st c1st 7676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-iota 6307  df-fv 6356
This theorem is referenced by:  filnetlem2  33624  filnetlem3  33625  filnetlem4  33626
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