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Theorem filnetlem1 32710
Description: Lemma for filnet 32714. 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 6332 . . . 4 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
21sseq2d 3782 . . 3 (𝑥 = 𝐴 → ((1st𝑦) ⊆ (1st𝑥) ↔ (1st𝑦) ⊆ (1st𝐴)))
3 fveq2 6332 . . . 4 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
43sseq1d 3781 . . 3 (𝑦 = 𝐵 → ((1st𝑦) ⊆ (1st𝐴) ↔ (1st𝐵) ⊆ (1st𝐴)))
52, 4sylan9bb 499 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → ((1st𝑦) ⊆ (1st𝑥) ↔ (1st𝐵) ⊆ (1st𝐴)))
6 filnet.d . 2 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}
75, 6brab2a 5334 1 (𝐴𝐷𝐵 ↔ ((𝐴𝐻𝐵𝐻) ∧ (1st𝐵) ⊆ (1st𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  wss 3723  {csn 4316   ciun 4654   class class class wbr 4786  {copab 4846   × cxp 5247  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-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  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-br 4787  df-opab 4847  df-xp 5255  df-iota 5994  df-fv 6039
This theorem is referenced by:  filnetlem2  32711  filnetlem3  32712  filnetlem4  32713
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