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Theorem 1stcclb 21228
Description: A property of points in a first-countable topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
Hypothesis
Ref Expression
1stcclb.1 𝑋 = 𝐽
Assertion
Ref Expression
1stcclb ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐽,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Proof of Theorem 1stcclb
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 1stcclb.1 . . . 4 𝑋 = 𝐽
21is1stc2 21226 . . 3 (𝐽 ∈ 1st𝜔 ↔ (𝐽 ∈ Top ∧ ∀𝑤𝑋𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦)))))
32simprbi 480 . 2 (𝐽 ∈ 1st𝜔 → ∀𝑤𝑋𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))))
4 eleq1 2687 . . . . . . 7 (𝑤 = 𝐴 → (𝑤𝑦𝐴𝑦))
5 eleq1 2687 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑤𝑧𝐴𝑧))
65anbi1d 740 . . . . . . . 8 (𝑤 = 𝐴 → ((𝑤𝑧𝑧𝑦) ↔ (𝐴𝑧𝑧𝑦)))
76rexbidv 3048 . . . . . . 7 (𝑤 = 𝐴 → (∃𝑧𝑥 (𝑤𝑧𝑧𝑦) ↔ ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))
84, 7imbi12d 334 . . . . . 6 (𝑤 = 𝐴 → ((𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦)) ↔ (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
98ralbidv 2983 . . . . 5 (𝑤 = 𝐴 → (∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦)) ↔ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
109anbi2d 739 . . . 4 (𝑤 = 𝐴 → ((𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))) ↔ (𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))))
1110rexbidv 3048 . . 3 (𝑤 = 𝐴 → (∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))) ↔ ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))))
1211rspcv 3300 . 2 (𝐴𝑋 → (∀𝑤𝑋𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))))
133, 12mpan9 486 1 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  wrex 2910  wss 3567  𝒫 cpw 4149   cuni 4427   class class class wbr 4644  ωcom 7050  cdom 7938  Topctop 20679  1st𝜔c1stc 21221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-in 3574  df-ss 3581  df-pw 4151  df-uni 4428  df-1stc 21223
This theorem is referenced by:  1stcfb  21229  1stcrest  21237  lly1stc  21280  tx1stc  21434
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